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CHAPTER 3 Ratio, Proportion, and Percent

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When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. A ratio compares two quantities that are measured with the same unit. If we compare $a$ and $b$ , the ratio is written as $a\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}$

3.1 Ratios and Rate

Learning Objectives

By the end of this section, you will be able to:

Write a ratio as a fraction
Find unit rates
Find unit price
Translate phrases to expressions with fractions

Write a Ratio as a Fraction

Ratios

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}$

In this section, we will use the fraction notation. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as $\frac{4}{1}$ instead of simplifying it to $4$ so that we can see the two parts of the ratio.

EXAMPLE 1

Write each ratio as a fraction: a) $\phantom{\rule{0.2em}{0ex}}15\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}27\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}45\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}18$ .

Solution

	$\text{15 to 27}$
Write as a fraction with the first number in the numerator and the second in the denominator.	$\frac{15}{27}$
Simplify the fraction.	$\frac{5}{9}$

We leave the ratio in b) as an improper fraction.

	$\text{45 to 18}$
Write as a fraction with the first number in the numerator and the second in the denominator.	$\frac{45}{18}$
Simplify.	$\frac{5}{2}$

TRY IT 1.1

Write each ratio as a fraction: a) $\phantom{\rule{0.2em}{0ex}}21\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}56\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}48\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}32$ .

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{3}{8}$
$\phantom{\rule{0.2em}{0ex}}\frac{3}{2}$

TRY IT 1.2

Write each ratio as a fraction: a) $\phantom{\rule{0.2em}{0ex}}27\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}72\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}51\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}34$ .

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{3}{8}$
$\phantom{\rule{0.2em}{0ex}}\frac{3}{2}$

Ratios Involving Decimals

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.

For example, consider the ratio $0.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.05$ . We can write it as a fraction with decimals and then multiply the numerator and denominator by $100$ to eliminate the decimals.

$A fraction is shown with 0.8 in the numerator and 0.05 in the denominator. Below it is the same fraction with both the numerator and denominator multiplied by 100. Below that is a fraction with 80 in the numerator and 5 in the denominator.$

Do you see a shortcut to find the equivalent fraction? Notice that $0.8=\frac{8}{10}$ and $0.05=\frac{5}{100}$ . The least common denominator of $\frac{8}{10}$ and $\frac{5}{100}$ is $100$ . By multiplying the numerator and denominator of $\frac{0.8}{0.05}$ by $100$ , we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this:


“Move” the decimal 2 places.	$\frac{80}{5}$
Simplify.	$\frac{16}{1}$

You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same.

EXAMPLE 2

Write each ratio as a fraction of whole numbers:

a) $\phantom{\rule{0.2em}{0ex}}4.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.2$

b) $\phantom{\rule{0.2em}{0ex}}2.7\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54$

Solution

a) $\phantom{\rule{0.2em}{0ex}}\text{4.8 to 11.2}$
Write as a fraction.	$\frac{4.8}{11.2}$
Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right.	$\frac{48}{112}$
Simplify.	$\frac{3}{7}$

So $4.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.2$ is equivalent to $\frac{3}{7}$ .

b) The numerator has one decimal place and the denominator has $2$ . To clear both decimals we need to move the decimal $2$ places to the right. $2.7\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54$
Write as a fraction.	$\frac{2.7}{0.54}$
Move both decimals right two places.	$\frac{270}{54}$
Simplify.	$\frac{5}{1}$

So $2.7\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54$ is equivalent to $\frac{5}{1}$ .

TRY IT 2.1

Write each ratio as a fraction: a) $\phantom{\rule{0.2em}{0ex}}4.6\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.5\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}2.3\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.69$ .

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{10}{3}$

TRY IT 2.2

Write each ratio as a fraction: a) $\phantom{\rule{0.2em}{0ex}}3.4\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}15.3\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}3.4\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.68$ .

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{2}{9}$
$\phantom{\rule{0.2em}{0ex}}\frac{5}{1}$

Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.

EXAMPLE 3

Write the ratio of $1\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{8}$ as a fraction.

Solution

	$1\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{8}$
Write as a fraction.	$\frac{1\frac{1}{4}}{2\frac{3}{8}}$
Convert the numerator and denominator to improper fractions.	$\frac{\frac{5}{4}}{\frac{19}{8}}$
Rewrite as a division of fractions.	$\frac{5}{4}\div\frac{19}{8}$
Invert the divisor and multiply.	$\frac{5}{4}\cdot\frac{8}{19}$
Simplify.	$\frac{10}{19}$

TRY IT 3.1

Write each ratio as a fraction: $1\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{5}{8}$ .

Answer

$\frac{2}{3}$

TRY IT 3.2

Write each ratio as a fraction: $1\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{4}$ .

Answer

$\frac{9}{22}$

Applications of Ratios

One real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person’s overall health. A ratio of less than $5$ to 1 is considered good.

EXAMPLE 4

Hector’s total cholesterol is $249$ mg/dl and his HDL cholesterol is $39$ mg/dl. a) Find the ratio of his total cholesterol to his HDL cholesterol. b) Assuming that a ratio less than $5$ to $1$ is considered good, what would you suggest to Hector?

Solution

a) First, write the words that express the ratio. We want to know the ratio of Hector’s total cholesterol to his HDL cholesterol.

Write as a fraction.	$\frac{\text{total cholesterol}}{\text{HDL cholesterol}}$
Substitute the values.	$\frac{249}{39}$
Simplify.	$\frac{83}{13}$

b) Is Hector’s cholesterol ratio ok? If we divide $83$ by $13$ we obtain approximately $6.4$ , so $\frac{83}{13}\approx \frac{6.4}{1}$ . Hector’s cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.

TRY IT 4.1

Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is $185$ mg/dL and HDL cholesterol is $40$ mg/dL.

Answer

$\frac{37}{8}$

TRY IT 4.2

Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is $204$ mg/dL and HDL cholesterol is $38$ mg/dL.

Answer

$\frac{102}{19}$

Ratios of Two Measurements in Different Units

To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units.

We know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit.

EXAMPLE 5

The Canadian National Building Code (CNBC) Guidelines for wheel chair ramps require a maximum vertical rise of $1$ inch for every $1$ foot of horizontal run. What is the ratio of the rise to the run?

Solution

In a ratio, the measurements must be in the same units. We can change feet to inches, or inches to feet. It is usually easier to convert to the smaller unit, since this avoids introducing more fractions into the problem.

Write the words that express the ratio.

	Ratio of the rise to the run
Write the ratio as a fraction.	$\frac{\text{rise}}{\text{run}}$
Substitute in the given values.	$\frac{\text{1 inch}}{\text{1 foot}}$
Convert 1 foot to inches.	$\frac{\text{1 inch}}{\text{12 inches}}$
Simplify, dividing out common factors and units.	$\frac{1}{12}$

So the ratio of rise to run is $1$ to $12$ . This means that the ramp should rise $1$ inch for every $12$ inches of horizontal run to comply with the guidelines.

TRY IT 5.1

Find the ratio of the first length to the second length: $32$ inches to $1$ foot.

Answer

$\frac{8}{3}$

TRY IT 5.2

Find the ratio of the first length to the second length: $1$ foot to $54$ inches.

Answer

$\frac{2}{9}$

Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are $120$ miles in $2$ hours, $160$ words in $4$ minutes, and $\text{\$5}$ dollars per $64$ ounces.

Rate

A rate compares two quantities of different units. A rate is usually written as a fraction.

When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

EXAMPLE 6

Bob drove his car $525$ miles in $9$ hours. Write this rate as a fraction.

Solution

	$\text{525 miles in 9 hours}$
Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator.	$\frac{\text{525 miles}}{\text{9 hours}}$
	$\frac{\text{175 miles}}{\text{3 hours}}$

So $525$ miles in $9$ hours is equivalent to $\frac{\text{175 miles}}{\text{3 hours}}$ .

TRY IT 6.1

Write the rate as a fraction: $492$ miles in $8$ hours.

Answer

$\frac{\text{123 miles}}{\text{2 hours}}$

TRY IT 6.2

Write the rate as a fraction: $242$ miles in $6$ hours.

Answer

$\frac{\text{121 miles}}{\text{3 hours}}$

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of $\frac{\text{175 miles}}{\text{3 hours}}$ . This tells us that every three hours, Bob will travel $175$ miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of $1$ unit is referred to as a unit rate.

Unit Rate

A unit rate is a rate with denominator of $1$ unit.

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of $68$ miles per hour we mean that we travel $68$ miles in $1$ hour. We would write this rate as $68$ miles/hour (read $68$ miles per hour). The common abbreviation for this is $68$ mph. Note that when no number is written before a unit, it is assumed to be $1$ .

So $68$ miles/hour really means $\text{68 miles/1 hour.}$

Two rates we often use when driving can be written in different forms, as shown:

Example	Rate	Write	Abbreviate	Read
$68$ miles in $1$ hour	$\frac{\text{68 miles}}{\text{1 hour}}$	$68$ miles/hour	$68$ mph	$\text{68 miles per hour}$
$36$ miles to $1$ gallon	$\frac{\text{36 miles}}{\text{1 gallon}}$	$36$ miles/gallon	$36$ mpg	$\text{36 miles per gallon}$

Example

Rate

Write

Abbreviate

Read

$68$ miles in $1$ hour

$\frac{\text{68 miles}}{\text{1 hour}}$

$68$ miles/hour

$68$ mph

$\text{68 miles per hour}$

$36$ miles to $1$ gallon

$\frac{\text{36 miles}}{\text{1 gallon}}$

$36$ miles/gallon

$36$ mpg

$\text{36 miles per gallon}$

Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid $\text{\$12.50}$ for each hour you work, you could write that your hourly (unit) pay rate is $\text{\$12.50/hour}$ (read $\text{\$12.50}$ per hour.)

To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of $1$ .

EXAMPLE 7

Anita was paid $\text{\$384}$ last week for working $\text{32 hours}$ . What is Anita’s hourly pay rate?

Solution

Start with a rate of dollars to hours. Then divide.	$\text{\$384 last week for 32 hours}$
Write as a rate.	$\frac{\$384}{\text{32 hours}}$
Divide the numerator by the denominator.	$\frac{\$12}{\text{1 hour}}$
Rewrite as a rate.	$\$12/\text{hour}$

Anita’s hourly pay rate is $\text{\$12}$ per hour.

TRY IT 7.1

Find the unit rate: $\text{\$630}$ for $35$ hours.

Answer

$18.00/hour

TRY IT 7.2

Find the unit rate: $\text{\$684}$ for $36$ hours.

Answer

$19.00/hour

EXAMPLE 8

Sven drives his car $455$ miles, using $14$ gallons of gasoline. How many miles per gallon does his car get?

Solution

Start with a rate of miles to gallons. Then divide.

	$\text{455 miles to 14 gallons of gas}$
Write as a rate.	$\frac{\text{455 miles}}{\text{14 gallons}}$
Divide 455 by 14 to get the unit rate.	$\frac{\text{32.5 miles}}{\text{1 gallon}}$

Sven’s car gets $32.5$ miles/gallon, or $32.5$ mpg.

TRY IT 8.1

Find the unit rate: $423$ miles to $18$ gallons of gas.

Answer

23.5 mpg

TRY IT 8.2

Find the unit rate: $406$ miles to $14.5$ gallons of gas.

Answer

28 mpg

Find Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item.

Unit price

A unit price is a unit rate that gives the price of one item.

EXAMPLE 9

The grocery store charges $\text{\$3.99}$ for a case of $24$ bottles of water. What is the unit price?

Solution

What are we asked to find? We are asked to find the unit price, which is the price per bottle.

Write as a rate.	$\frac{\$3.99}{\text{24 bottles}}$
Divide to find the unit price.	$\frac{\$0.16625}{\text{1 bottle}}$
Round the result to the nearest penny.	$\frac{\$0.17}{\text{1 bottle}}$

The unit price is approximately $\text{\$0.17}$ per bottle. Each bottle costs about $\text{\$0.17}$ .

TRY IT 9.1

Find the unit price. Round your answer to the nearest cent if necessary.

$\text{24-pack}$ of juice boxes for $\text{\$6.99}$

Answer

$0.29/box

TRY IT 9.2

Find the unit price. Round your answer to the nearest cent if necessary.

$\text{24-pack}$ of bottles of ice tea for $\text{\$12.72}$

Answer

$0.53/bottle

Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.

EXAMPLE 10

Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at $\text{\$14.99}$ for $64$ loads of laundry and the same brand of powder detergent is priced at $\text{\$15.99}$ for $80$ loads.

Which is the better buy, the liquid or the powder detergent?

Solution

To compare the prices, we first find the unit price for each type of detergent.

	Liquid	Powder
Write as a rate.	$\frac{\text{\$14.99}}{\text{64 loads}}$	$\frac{\text{\$15.99}}{\text{80 loads}}$
Find the unit price.	$\frac{\text{\$0.234…}}{\text{1 load}}$	$\frac{\text{\$0.199…}}{\text{1 load}}$
Round to the nearest cent.	$\begin{array}{c}\text{\$0.23/load}\hfill \\ \text{(23 cents per load.)}\hfill \end{array}$	$\begin{array}{c}\text{\$0.20/load}\hfill \\ \text{(20 cents per load)}\hfill \end{array}$

Now we compare the unit prices. The unit price of the liquid detergent is about $\text{\$0.23}$ per load and the unit price of the powder detergent is about $\text{\$0.20}$ per load. The powder is the better buy.

TRY IT 10.1

Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand A Storage Bags, $\text{\$4.59}$ for $40$ count, or Brand B Storage Bags, $\text{\$3.99}$ for $30$ count

Answer

Brand A costs $0.11 per bag. Brand B costs $0.13 per bag. Brand A is the better buy.

TRY IT 10.2

Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand C Chicken Noodle Soup, $\text{\$1.89}$ for $26$ ounces, or Brand D Chicken Noodle Soup, $\text{\$0.95}$ for $10.75$ ounces

Answer

Brand C costs $0.07 per ounce. Brand D costs $0.09 per ounce. Brand C is the better buy.

Notice in the above example that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.

Translate Phrases to Expressions with Fractions

Have you noticed that the examples in this section used the comparison words ratio of, to, per, in, for, on, and from? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.

EXAMPLE 11

Translate the word phrase into an algebraic expression:

a) $\phantom{\rule{0.2em}{0ex}}427$ miles per $h$ hours

b) $\phantom{\rule{0.2em}{0ex}}x$ students to $3$ teachers

c) $\phantom{\rule{0.2em}{0ex}}y$ dollars for $18$ hours

Solution

a)
	$\text{427 miles per}\phantom{\rule{0.2em}{0ex}}h\phantom{\rule{0.2em}{0ex}}\text{hours}$
Write as a rate.	$\frac{\text{427 miles}}{h\phantom{\rule{0.2em}{0ex}}\text{hours}}$

b)
	$x\phantom{\rule{0.2em}{0ex}}\text{students to 3 teachers}$
Write as a rate.	$\frac{x\phantom{\rule{0.2em}{0ex}}\text{students}}{\text{3 teachers}}$

c)
	$y\phantom{\rule{0.2em}{0ex}}\text{dollars for 18 hours}$
Write as a rate.	$\frac{\$y}{\text{18 hours}}$

TRY IT 11.1

Translate the word phrase into an algebraic expression.

a) $\phantom{\rule{0.2em}{0ex}}689$ miles per $h$ hours b) $y$ parents to $22$ students c) $d$ dollars for $9$ minutes

Answer

689 mi/h hours
y parents/22 students
$d/9 min

TRY IT 11.2

Translate the word phrase into an algebraic expression.

a) $m$ miles per $9$ hours b) $x$ students to $8$ buses c) $y$ dollars for $40$ hours

Answer

m mi/9 h
x students/8 buses
$y/40 h

Access to Additional Online R

Glossary

ratio: A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a$ to $b$ , $\frac{a}{b}$ , or $a:b$ .

rate: A rate compares two quantities of different units. A rate is usually written as a fraction.

unit rate: A unit rate is a rate with denominator of 1 unit.

unit price: A unit price is a unit rate that gives the price of one item.

Practice Makes Perfect

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction.

1. $20$ to $36$	2. $20$ to $32$
3. $42$ to $48$	4. $45$ to $54$
5. $49$ to $21$	6. $56$ to $16$
7. $84$ to $36$	8. $6.4$ to $0.8$
9. $0.56$ to $2.8$	10. $1.26$ to $4.2$
11. $1\frac{2}{3}$ to $2\frac{5}{6}$	12. $1\frac{3}{4}$ to $2\frac{5}{8}$
13. $4\frac{1}{6}$ to $3\frac{1}{3}$	14. $5\frac{3}{5}$ to $3\frac{3}{5}$
15. $\text{\$18}$ to $\text{\$63}$	16. $\text{\$16}$ to $\text{\$72}$
17. $\text{\$1.21}$ to $\text{\$0.44}$	18. $\text{\$1.38}$ to $\text{\$0.69}$
19. $28$ ounces to $84$ ounces	20. $32$ ounces to $128$ ounces
21. $12$ feet to $46$ feet	22. $15$ feet to $57$ feet
23. $246$ milligrams to $45$ milligrams	24. $304$ milligrams to $48$ milligrams
25. total cholesterol of $175$ to HDL cholesterol of $45$	26. total cholesterol of $215$ to HDL cholesterol of $55$
27. $27$ inches to $1$ foot	28. $28$ inches to $1$ foot

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction.

29. $140$ calories per $12$ ounces	30. $180$ calories per $16$ ounces
31. $8.2$ pounds per $3$ square inches	32. $9.5$ pounds per $4$ square inches
33. $488$ miles in $7$ hours	34. $527$ miles in $9$ hours
35. $\text{\$595}$ for $40$ hours	36. $\text{\$798}$ for $40$ hours

Find Unit Rates

In the following exercises, find the unit rate. Round to two decimal places, if necessary.

37. $140$ calories per $12$ ounces	38. $180$ calories per $16$ ounces
39. $8.2$ pounds per $3$ square inches	40. $9.5$ pounds per $4$ square inches
41. $488$ miles in $7$ hours	42. $527$ miles in $9$ hours
43. $\text{\$595}$ for $40$ hours	44. $\text{\$798}$ for $40$ hours
45. $576$ miles on $18$ gallons of gas	46. $435$ miles on $15$ gallons of gas
47. $43$ pounds in $16$ weeks	48. $57$ pounds in $24$ weeks
49. $46$ beats in $0.5$ minute	50. $54$ beats in $0.5$ minute
51. The bindery at a printing plant assembles $96,000$ magazines in $12$ hours. How many magazines are assembled in one hour?	52. The pressroom at a printing plant prints $540,000$ sections in $12$ hours. How many sections are printed per hour?

Find Unit Price

In the following exercises, find the unit price. Round to the nearest cent.

53. Soap bars at $8$ for $\text{\$8.69}$	54. Soap bars at $4$ for $\text{\$3.39}$
55. Women’s sports socks at $6$ pairs for $\text{\$7.99}$	56. Men’s dress socks at $3$ pairs for $\text{\$8.49}$
57. Snack packs of cookies at $12$ for $\text{\$5.79}$	58. Granola bars at $5$ for $\text{\$3.69}$
59. CD-RW discs at $25$ for $\text{\$14.99}$	60. CDs at $50$ for $\text{\$4.49}$
61. The grocery store has a special on macaroni and cheese. The price is $\text{\$3.87}$ for $3$ boxes. How much does each box cost?	62. The pet store has a special on cat food. The price is $\text{\$4.32}$ for $12$ cans. How much does each can cost?

In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.

63. Mouthwash, $\text{50.7-ounce}$ size for $\text{\$6.99}$ or $\text{33.8-ounce}$ size for $\text{\$4.79}$	64. Toothpaste, $6$ ounce size for $\text{\$3.19}$ or $7.8-ounce$ size for $\text{\$5.19}$
65. Breakfast cereal, $18$ ounces for $\text{\$3.99}$ or $14$ ounces for $\text{\$3.29}$	66. Breakfast Cereal, $10.7$ ounces for $\text{\$2.69}$ or $14.8$ ounces for $\text{\$3.69}$
67. Ketchup, $\text{40-ounce}$ regular bottle for $\text{\$2.99}$ or $\text{64-ounce}$ squeeze bottle for $\text{\$4.39}$	68. Mayonnaise $\text{15-ounce}$ regular bottle for $\text{\$3.49}$ or $\text{22-ounce}$ squeeze bottle for $\text{\$4.99}$
69. Cheese $\text{\$6.49}$ for $1$ lb. block or $\text{\$3.39}$ for $\frac{1}{2}$ lb. block	70. Candy $\text{\$10.99}$ for a $1$ lb. bag or $\text{\$2.89}$ for $\frac{1}{4}$ lb. of loose candy

Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.

71. $793$ miles per $p$ hours	72. $78$ feet per $r$ seconds
73. $\text{\$3}$ for $0.5$ lbs.	74. $j$ beats in $0.5$ minutes
75. 105 calories in $x$ ounces	76. $400$ minutes for $m$ dollars
77. the ratio of $y$ and $5x$	78. the ratio of $12x$ and $y$

Everyday Math

79. One elementary school in Saskatchewan has $684$ students and $45$ teachers. Write the student-to-teacher ratio as a unit rate.	80. The average Canadian produces about $350$ pounds of paper trash per year $\text{(365 days).}$ How many pounds of paper trash does the average Canadian produce each day? (Round to the nearest tenth of a pound.)
81. A popular fast food burger weighs $7.5$ ounces and contains $540$ calories, $29$ grams of fat, $43$ grams of carbohydrates, and $25$ grams of protein. Find the unit rate of a) calories per ounce b) grams of fat per ounce c) grams of carbohydrates per ounce d) grams of protein per ounce. Round to two decimal places.	82. A $16-ounce$ chocolate mocha coffee with whipped cream contains $470$ calories, $18$ grams of fat, $63$ grams of carbohydrates, and $15$ grams of protein. Find the unit rate of a) calories per ounce b) grams of fat per ounce c) grams of carbohydrates per ounce d) grams of protein per ounce.

Writing Exercises

83. Would you prefer the ratio of your income to your friend’s income to be $\text{3/1}$ or $1/3?$ Explain your reasoning.	84. The parking lot at the airport charges $\text{\$0.75}$ for every $15$ minutes. a) How much does it cost to park for $1$ hour? b) Explain how you got your answer to part a). Was your reasoning based on the unit cost or did you use another method?
85. Kathryn ate a $4-ounce$ cup of frozen yogurt and then went for a swim. The frozen yogurt had $115$ calories. Swimming burns $422$ calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.	86. Arjun had a $16-ounce$ cappuccino at his neighbourhood coffee shop. The cappuccino had $110$ calories. If Arjun walks for one hour, he burns $246$ calories. For how many minutes must Arjun walk to burn off the calories in the cappuccino? Explain your reasoning.

Answers

1. $\frac{5}{9}$	3. $\frac{7}{8}$	5. $\frac{7}{3}$
7. $\frac{7}{3}$	9. $\frac{1}{5}$	11. $\frac{10}{17}$
13. $\frac{5}{4}$	15. $\frac{2}{7}$	17. $\frac{11}{4}$
19. $\frac{1}{3}$	21. $\frac{6}{23}$	23. $\frac{82}{15}$
25. $\frac{35}{9}$	27. $\frac{9}{4}$	29. $\frac{\text{35 calories}}{\text{3 ounces}}$
31. $\frac{\text{41 lbs}}{\text{15 sq. in}.}$	33. $\frac{\text{488 miles}}{\text{7 hours}}$	35. $\frac{\text{\$119}}{\text{8 hours}}$
37. 11.67 calories/ounce	39. 2.73 lbs./sq. in.	41. 69.71 mph
43. $14.88/hour	45. 32 mpg	47. 2.69 lbs./week
49. 92 beats/minute	51. 8,000	53. $1.09/bar
55. $1.33/pair	57. $0.48/pack	59. $0.60/disc
61. $1.29/box	63. The 50.7-ounce size costs $0.138 per ounce. The 33.8-ounce size costs $0.142 per ounce. The 50.7-ounce size is the better buy.	65. The 18-ounce size costs $0.222 per ounce. The 14-ounce size costs $0.235 per ounce. The 18-ounce size is a better buy.
67. The regular bottle costs $0.075 per ounce. The squeeze bottle costs $0.069 per ounce. The squeeze bottle is a better buy.	69. The half-pound block costs $6.78/lb, so the 1-lb. block is a better buy.	71. $\frac{\text{793 miles}}{p\phantom{\rule{0.2em}{0ex}}\text{hours}}$
73. $\frac{\text{?3}}{\text{0.5 lbs}.}$	75. $\frac{\text{105 calories}}{x\phantom{\rule{0.2em}{0ex}}\text{ounces}}$	77. $\frac{y}{5x}$
79. 15.2 students per teacher	81. a) 72 calories/ounce b) 3.87 grams of fat/ounce c) 5.73 grams carbs/once d) 3.33 grams protein/ounce	83. Answers will vary.
85. Answers will vary.

Attributions

This chapter has been adapted from “Ratios and Rate” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

3.2 Understand Percent

Learning Objectives

By the end of this section, you will be able to:

Use the definition of percent
Convert percents to fractions and decimals
Convert decimals and fractions to percents

Use the Definition of Percent

How many cents are in one dollar? There are $100$ cents in a dollar. How many years are in a century? There are $100$ years in a century. Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A percent is a ratio whose denominator is $100$ . We use the percent symbol $\text{\%,}$ to show percent.

Percent

A percent is a ratio whose denominator is $100$ .

According to data from the Statistics Canada, $\text{57\%}$ of Canadian Internet users reported a cyber security incident, including being redirected to fraudulent websites that asked for personal information or getting a virus or other computer infection. This means $57$ out of every $100$ Canadian internet users reported cyber security incidents as (Figure 1) shows. Out of the $100$ squares on the grid, $57$ are shaded, which we write as the ratio $\frac{57}{100}$ .

The figure shows a hundred flat with 57 units shaded.

Figure 1

Similarly, $\text{25\%}$ means a ratio of $\frac{25}{100},\text{3\%}$ means a ratio of $\frac{3}{100}$ and $\text{100\%}$ means a ratio of $\frac{100}{100}$ . In words, “one hundred percent” means the total $\text{100\%}$ is $\frac{100}{100}$ , and since $\frac{100}{100}=1$ , we see that $\text{100\%}$ means $1$ whole.

EXAMPLE 1

According to a survey done by Universities Canada $\left(2017\right)\text{,}\phantom{\rule{0.2em}{0ex}}\text{71\%}$ of Canada’s Universities are working to include Indigenous representation within their governance or leadership structures.Write this percent as a ratio.

Solution

The amount we want to convert is 71%.	$71\%$
Write the percent as a ratio. Remember that percent means per 100.	$\frac{71}{100}$

TRY IT 1.1

Write the percent as a ratio.

According to a survey, $\text{89\%}$ of college students have a smartphone.

Answer

$\frac{89}{100}$

TRY IT 1.2

Write the percent as a ratio.

A study found that $\text{72\%}$ of Canadian teens send text messages regularly.

Answer

$\frac{72}{100}$

EXAMPLE 2

In $2018$ , according to a Universities Canada survey, $56$ out of every $100$ of today’s undergraduates benefit from experiential learning such as co-ops, internships and service learning. Write this as a ratio and then as a percent.

Solution

The amount we want to convert is $56$ out of $100$ .	$56$ out of $100$
Write as a ratio.	$\frac{56}{100}$
Convert the 56 per 100 to percent.	$56\%$

TRY IT 2.1

Write as a ratio and then as a percent: According to Statistics Canada, only $10$ out of $100$ young Canadians cross a provincial border to complete their university degree.

Answer

$\frac{10}{100},\text{10\%}$

TRY IT 2.2

Write as a ratio and then as a percent: According to an international comparison done by the British Council, $55$ out of $100$ current professional leaders across 30 countries and in all sectors, are liberal arts grads with bachelor’s degrees in the social sciences or humanities.

Answer

$\frac{55}{100},\text{55\%}$

Convert Percents to Fractions and Decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per $100$ , so the denominator of the fraction is $100$ .

Convert a Percent to a Fraction.

Write the percent as a ratio with the denominator $100$ .
Simplify the fraction if possible.

EXAMPLE 3

Convert each percent to a fraction:

$\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\text{36\%}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\text{125\%}$

Solution

a)
	$36\%$
Write as a ratio with denominator 100.	$\frac{36}{100}$
Simplify.	$\frac{9}{25}$

b)
	$125\%$
Write as a ratio with denominator 100.	$\frac{125}{100}$
Simplify.	$\frac{5}{4}$

TRY IT 3.1

Convert each percent to a fraction:

$\phantom{\rule{0.2em}{0ex}}\text{48\%}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\text{110\%}$

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{12}{25}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{11}{10}$

TRY IT 3.2

Convert each percent to a fraction:

$\phantom{\rule{0.2em}{0ex}}\text{64\%}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\text{150\%}$

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{16}{25}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{3}{2}$

The previous example shows that a percent can be greater than $1$ . We saw that $\text{125\%}$ means $\frac{125}{100}$ , or $\frac{5}{4}$ . These are improper fractions, and their values are greater than one.

EXAMPLE 4

Convert each percent to a fraction:

$\phantom{\rule{0.2em}{0ex}}\text{24.5\%}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}33\frac{1}{3}\%$

Solution

a)
	$24.5\%$
Write as a ratio with denominator 100.	$\frac{24.5}{100}$
Clear the decimal by multiplying numerator and denominator by 10.	$\frac{24.5\left(10\right)}{100\left(10\right)}$
Multiply.	$\frac{245}{1000}$
Rewrite showing common factors.	$\frac{5\cdot49}{5\cdot200}$
Simplify.	$\frac{49}{200}$

b)
	$33\frac{1}{3}\%$
Write as a ratio with denominator 100.	$\frac{33\frac{1}{3}}{100}$
Write the numerator as an improper fraction.	$\frac{\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\frac{100}{3}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}}{100}$
Rewrite as fraction division, replacing 100 with $\frac{100}{1}$ .	$\frac{100}{3}\div\frac{100}{1}$
Multiply by the reciprocal.	$\frac{100}{3}\cdot \frac{1}{100}$
Simplify.	$\frac{1}{3}$

TRY IT 4.1

Convert each percent to a fraction:

$\phantom{\rule{0.2em}{0ex}}\text{64.4\%}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}66\frac{2}{3}\%$

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{161}{250}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$

TRY IT 4.2

Convert each percent to a fraction:

$\phantom{\rule{0.2em}{0ex}}\text{42.5\%}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}8\frac{3}{4}\%$

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{17}{40}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{7}{80}$

To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

HOW TO: Convert a Percent to a Decimal

Write the percent as a ratio with the denominator $100$ .
Convert the fraction to a decimal by dividing the numerator by the denominator.

EXAMPLE 5

Convert each percent to a decimal:

$\phantom{\rule{0.4em}{0ex}}\text{6\%}\phantom{\rule{0.4em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\text{78\%}$

Solution

Because we want to change to a decimal, we will leave the fractions with denominator $100$ instead of removing common factors.

a)
	$6\%$
Write as a ratio with denominator 100.	$\frac{6}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.06$

b)
	$78\%$
Write as a ratio with denominator 100.	$\frac{78}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.78$

TRY IT 5.1

Convert each percent to a decimal:

$\phantom{\rule{0.3em}{0ex}}\text{9\%}\phantom{\rule{0.3em}{0ex}}$
$\phantom{\rule{0.3em}{0ex}}\text{87\%}$

Answer

0.09
0.87

TRY IT 5.2

Convert each percent to a decimal:

$\phantom{\rule{0.3em}{0ex}}\text{3\%}\phantom{\rule{0.3em}{0ex}}$
$\phantom{\rule{0.3em}{0ex}}\text{91\%}$

Answer

0.03
0.91

EXAMPLE 6

Convert each percent to a decimal:

$\phantom{\rule{0.3em}{0ex}}\text{135\%}\phantom{\rule{0.3em}{0ex}}$
$\phantom{\rule{0.3em}{0ex}}\text{12.5\%}$

Solution

a)
	$135\%$
Write as a ratio with denominator 100.	$\frac{135}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$1.35$

b)
	$12.5\%$
Write as a ratio with denominator 100.	$\frac{12.5}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.125$

TRY IT 6.1

Convert each percent to a decimal:

$\phantom{\rule{0.3em}{0ex}}\text{115\%}\phantom{\rule{0.3em}{0ex}}$
$\phantom{\rule{0.3em}{0ex}}\text{23.5\%}$

Answer

1.15
0.235

TRY IT 6.2

Convert each percent to a decimal:

$\phantom{\rule{0.3em}{0ex}}\text{123\%}\phantom{\rule{0.3em}{0ex}}$
$\phantom{\rule{0.3em}{0ex}}\text{16.8\%}$

Answer

1.23
0.168

Let’s summarize the results from the previous examples in the table below, and look for a pattern we could use to quickly convert a percent number to a decimal number.

Percent	Decimal
$\text{6\%}$	$0.06$
$\text{78\%}$	$0.78$
$\text{135\%}$	$1.35$
$\text{12.5\%}$	$0.125$

Do you see the pattern?

To convert a percent number to a decimal number, we move the decimal point two places to the left and remove the $\%$ sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer $6$ as $6.0$ , we can think of $\text{6\%}$ as $\text{6.0\%}$ .) Notice that we may need to add zeros in front of the number when moving the decimal to the left.

(Figure 2) uses the percents in the table above and shows visually how to convert them to decimals by moving the decimal point two places to the left.

The figures shows two columns and five rows . The first row is a header row and it labels each column “Percent” and “Decimal”. Under the “Percent” column are the values: 6%, 78%, 135%, 12.5%. Under the “Decimal” column are the values: 0.06, 0.78, 1.35, 0.125. There are two jumps for each percent to show how to convert it to a decimal.

Figure 2

EXAMPLE 7

Among a group of business leaders, $\text{77\%}$ believe that poor math and science education in Canada will lead to higher unemployment rates.

Convert the percent to: a) a fraction b) a decimal

Solution

a)
	$77\%$
Write as a ratio with denominator 100.	$\frac{77}{100}$

b)
	$\frac{77}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.77$

TRY IT 7.1

Convert the percent to: a) a fraction and b) a decimal

Twitter’s share of web traffic jumped $\text{24\%}$ when one celebrity tweeted live on air.

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{6}{25}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}0.24$

TRY IT 7.2

Convert the percent to: a) a fraction and b) a decimal

Statistics Canada shows that in $2016,\text{29\%}$ of adults aged $25$ to $64$ had a bachelor degree.

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{29}{100}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}0.29$

EXAMPLE 8

There are four suits of cards in a deck of cards—hearts, diamonds, clubs, and spades. The probability of randomly choosing a heart from a shuffled deck of cards is $\text{25\%}$ . Convert the percent to:

a fraction
a decimal

The figure shows someone holding a deck of cards.

(credit: Riles32807, Wikimedia Commons)

Solution

a)
	$25\%$
Write as a ratio with denominator 100.	$\frac{25}{100}$
Simplify.	$\frac{1}{4}$

b)	$\frac{1}{4}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.25$

TRY IT 8.1

Convert the percent to: a) a fraction, and b) a decimal

The probability that it will rain Monday is $\text{30\%}$ .

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{3}{10}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}0.3$

TRY IT 8.2

Convert the percent to: a) a fraction, and b) a decimal

The probability of getting heads three times when tossing a coin three times is $\text{12.5\%}$ .

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{12.5}{100}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}0.125$

Convert Decimals and Fractions to Percents

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is $100$ , it is easy to change that fraction to a percent.

HOW TO: Convert a Decimal to a Percent

Write the decimal as a fraction.
If the denominator of the fraction is not $100$ , rewrite it as an equivalent fraction with denominator $100$ .
Write this ratio as a percent.

EXAMPLE 9

Convert each decimal to a percent: a) $\phantom{\rule{0.2em}{0ex}}0.05\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.83$

Solution

a)
	$0.05$
Write as a fraction. The denominator is 100.	$\frac{5}{100}$
Write this ratio as a percent.	$5\%$

b)
	$0.83$
The denominator is 100.	$\frac{83}{100}$
Write this ratio as a percent.	$83\%$

TRY IT 9.1

Convert each decimal to a percent: a) $\phantom{\rule{0.2em}{0ex}}0.01\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.17$ .

Answer

TRY IT 9.2

Convert each decimal to a percent: a) $\phantom{\rule{0.2em}{0ex}}0.04\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.41$

Answer

To convert a mixed number to a percent, we first write it as an improper fraction.

EXAMPLE 10

Convert each decimal to a percent: a) $\phantom{\rule{0.2em}{0ex}}1.05\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.075$

Solution

a)
	$1.05$
Write as a fraction.	$1\frac{5}{100}$
Write as an improper fraction. The denominator is 100.	$\frac{105}{100}$
Write this ratio as a percent.	$105\%$

Notice that since $1.05$ > $1$ , the result is more than $\text{100\%.}$

b)
	$0.075$
Write as a fraction. The denominator is 1,000.	$\frac{75}{1,000}$
Divide the numerator and denominator by 10, so that the denominator is 100.	$\frac{7.5}{100}$
Write this ratio as a percent.	$7.5\%$

TRY IT 10.1

Convert each decimal to a percent: a) $\phantom{\rule{0.2em}{0ex}}1.75\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.0825$

Answer

175%
8.25%

TRY IT 10.2

Convert each decimal to a percent: a) $\phantom{\rule{0.2em}{0ex}}2.25\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.0925$

Answer

225%
9.25%

Let’s summarize the results from the previous examples in the table below so we can look for a pattern.

Decimal	Percent
$0.05$	$\text{5\%}$
$0.83$	$\text{83\%}$
$1.05$	$\text{105\%}$
$0.075$	$\text{7.5\%}$

Do you see the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

(Figure.3) uses the decimal numbers in the table above and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the $\%$ sign.

The figure shows two columns and five rows. The first row is a header row and it labels each column “Decimal” and “Percent”. Under the “Decimal” column are the values: 0.05, 0.83, 1.05, 0.075, 0.3. Under the “Percent” column are the values: 5%, 83%, 105%, 7.5%, 30%. There are two jumps for each decimal to show how to convert it to a percent.

Figure. 3

Now we also know how to change decimals to percents. So to convert a fraction to a percent, we first change it to a decimal and then convert that decimal to a percent.

HOW TO: Convert a Fraction to a Percent

Convert the fraction to a decimal.
Convert the decimal to a percent.

EXAMPLE 11

Convert each fraction or mixed number to a percent: a) $\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}\frac{11}{8}\phantom{\rule{0.2em}{0ex}}$ c) $\phantom{\rule{0.2em}{0ex}}2\frac{1}{5}$

Solution

To convert a fraction to a decimal, divide the numerator by the denominator.

a)
Change to a decimal.	$\frac{3}{4}$
Write as a percent by moving the decimal two places.
	$75\%$

b)
Change to a decimal.	$\frac{11}{8}$
Write as a percent by moving the decimal two places.
	$137.5\%$

c)
Write as an improper fraction.	$2\frac{1}{5}$
Change to a decimal.	$\frac{11}{5}$
Write as a percent.
	$220\%$

Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.

TRY IT 11.1

Convert each fraction or mixed number to a percent: a) $\frac{5}{8}\phantom{\rule{0.2em}{0ex}}$ b) $\frac{11}{4}\phantom{\rule{0.2em}{0ex}}$ c) $3\frac{2}{5}$

Answer

62.5%
275%
340%

TRY IT 11.2

Convert each fraction or mixed number to a percent: a) $\phantom{\rule{0.2em}{0ex}}\frac{7}{8}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}\frac{9}{4}\phantom{\rule{0.2em}{0ex}}$ c) $\phantom{\rule{0.2em}{0ex}}1\frac{3}{5}$

Answer

87.5%
225%
160%

Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.

EXAMPLE 12

Convert $\frac{5}{7}$ to a percent.

Solution

To change a fraction to a decimal, we divide the numerator by the denominator.

	$\frac{5}{7}$
Change to a decimal—rounding to the nearest thousandth.	$0.714$
Write as a percent.	$71.4\%$

TRY IT 12.1

Convert the fraction to a percent: $\frac{3}{7}$

Answer

42.9%

TRY IT 12.2

Convert the fraction to a percent: $\frac{4}{7}$

Answer

57.1%

When we first looked at fractions and decimals, we saw that fractions converted to a repeating decimal. When we converted the fraction $\frac{4}{3}$ to a decimal, we wrote the answer as $1.\overline{3}$ . We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

EXAMPLE 13

Statistics Canada reported in 2018 that approximately $\frac{1}{3}$ of Canadian adults are obese. Convert the fraction $\frac{1}{3}$ to a percent.

Solution

	$\frac{1}{3}$
Change to a decimal.
Write as a repeating decimal.	$0.333\dots$
Write as a percent.	$33\frac{1}{3}\%$

We could also write the percent as $33.\stackrel{-}{3}\%$ .

TRY IT 13.1

Convert the fraction to a percent:

According to the Canadian Census 2016, about $\frac{33}{50}$ people within the population of Canada are between the ages of $15$ and $64$ .

Answer

$66\%$

TRY IT 13.2

Convert the fraction to a percent:

According to the Canadian Census 2015, about $\frac{1}{6}$ of Canadian residents under age 18 are low income.

Answer

$16.\stackrel{-}{6}\text{\%,}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}16\frac{2}{3}\%$

Key Concepts

Convert a percent to a fraction.
1. Write the percent as a ratio with the denominator 100.
2. Simplify the fraction if possible.
Convert a percent to a decimal.
1. Write the percent as a ratio with the denominator 100.
2. Convert the fraction to a decimal by dividing the numerator by the denominator.
Convert a decimal to a percent.
1. Write the decimal as a fraction.
2. If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100.
3. Write this ratio as a percent.
Convert a fraction to a percent.
1. Convert the fraction to a decimal.
2. Convert the decimal to a percent.

Glossary

percent: A percent is a ratio whose denominator is $100$ .

Practice Makes Perfect

Use the Definition of Percents

In the following exercises, write each percent as a ratio.

1. In $2014$ , the unemployment rate for those with only a high school degree was $\text{6.0\%}$ .	2. In $2015$ , among the unemployed, $\text{29\%}$ were long-term unemployed.
3. The unemployment rate for those with Bachelor’s degrees was $\text{3.2\%}$ in $2014$ .	4. The unemployment rate in Canada in $2019$ was $\text{13.7\%}$ . In the following exercises, write as a) a ratio and b) a percent
5. $57$ out of $100$ nursing candidates received their degree at a community college.	6. $80$ out of $100$ firefighters and law enforcement officers were educated at a community college.
7. $42$ out of $100$ first-time freshmen students attend a community college.	8. $71$ out of $100$ full-time community college faculty have a master’s degree.

Convert Percents to Fractions and Decimals

In the following exercises, convert each percent to a fraction and simplify all fractions.

9. $\text{4\%}$	10. $\text{8\%}$
11. $\text{17\%}$	12. $\text{19\%}$
13. $\text{52\%}$	14. $\text{78\%}$
15. $\text{125\%}$	16. $\text{135\%}$
17. $\text{37.5\%}$	18. $\text{42.5\%}$
19. $\text{18.4\%}$	20. $\text{46.4\%}$
21. $9\frac{1}{2}\%$	22. $8\frac{1}{2}\%$
23. $5\frac{1}{3}\%$	24. $6\frac{2}{3}\%$

In the following exercises, convert each percent to a decimal.

25. $\text{5\%}$	26. $\text{9\%}$
27. $\text{1\%}$	28. $\text{2\%}$
29. $\text{63\%}$	30. $\text{71\%}$
31. $\text{40\%}$	32. $\text{50\%}$
33. $\text{115\%}$	34. $\text{125\%}$
35. $\text{150\%}$	36. $\text{250\%}$
37. $\text{21.4\%}$	38. $\text{39.3\%}$
39. $\text{7.8\%}$	40. $\text{6.4\%}$

In the following exercises, convert each percent to

a) a simplified fraction and

b) a decimal

41. In $2010,\text{1.5\%}$ of home sales had owner financing. (Source: Bloomberg Businessweek, 5/23–29/2011)	42. In $2016,\text{22.3\%}$ of the Canadian population was a visible minority. (Source: www12.statcan.gc.ca)
43. According to government data, in $2013$ the number of cell phones in India was $\text{70.23\%}$ of the population.	44. According to the Survey of Earned Doctorates, among Canadians age $25$ or older who had doctorate degrees in $2006,\text{44\%}$ are women.
45. A couple plans to have two children. The probability they will have two girls is $\text{25\%}$ .	46. Javier will choose one digit at random from $0$ through $9$ . The probability he will choose $3$ is $\text{10\%}$ .
47. According to the local weather report, the probability of thunderstorms in New York City on July $15$ is $\text{60\%}$ .	48. A club sells $50$ tickets to a raffle. Osbaldo bought one ticket. The probability he will win the raffle is $\text{2\%}$ .

Convert Decimals and Fractions to Percents

In the following exercises, convert each decimal to a percent.

49. $0.01$	50. $0.03$
51. $0.18$	52. $0.15$
53. $1.35$	54. $1.56$
55. $3$	56. $4$
57. $0.009$	58. $0.008$
59. $0.0875$	60. $0.0625$
61. $1.5$	62. $2.2$
63. $2.254$	64. $2.317$

In the following exercises, convert each fraction to a percent.

65. $\frac{1}{4}$	66. $\frac{1}{5}$
67. $\frac{3}{8}$	68. $\frac{5}{8}$
69. $\frac{7}{4}$	70. $\frac{9}{8}$
71. $6\frac{4}{5}$	72. $5\frac{1}{4}$
73. $\frac{5}{12}$	74. $\frac{11}{12}$
75. $2\frac{2}{3}$	76. $1\frac{2}{3}$
77. $\frac{3}{7}$	78. $\frac{6}{7}$
79. $\frac{5}{9}$	80. $\frac{4}{9}$

In the following exercises, convert each fraction to a percent.

81. $\frac{1}{4}$ of washing machines needed repair.

82. $\frac{1}{5}$ of dishwashers needed repair.

In the following exercises, convert each fraction to a percent.

83. According to the Government of Canada, in $2017,\frac{16}{25}$ of Canadian adults were overweight or obese.

84. Statistics Canada showed that in $2016,\text{15.4\%}$ of Canadian workers are using more than one language at work.

In the following exercises, complete the table.

85.

Fraction	Decimal	Percent
$\frac{1}{2}$
	$0.45$
		$18\%$
$\frac{1}{3}$
	$0.0008$
$2$

86.

Fraction	Decimal	Percent
$\frac{1}{4}$
	$0.65$
		$22\%$
$\frac{2}{3}$
	$0.0004$
$3$

Writing Exercises

87. Convert $\text{25\%},\text{50\%},\text{75\%},\text{and}\phantom{\rule{0.2em}{0ex}}\text{100\%}$ to fractions. Do you notice a pattern? Explain what the pattern is.	88. Convert $\frac{1}{10},\frac{2}{10},\frac{3}{10},\frac{4}{10},\frac{5}{10},\frac{6}{10},\frac{7}{10},\frac{8}{10}$ , and $\frac{9}{10}$ to percents. Do you notice a pattern? Explain what the pattern is.
89. When the Szetos sold their home, the selling price was $\text{500\%}$ of what they had paid for the house $\text{30 years}$ ago. Explain what $\text{500\%}$ means in this context.	90. According to cnn.com, cell phone use in $2008$ was $\text{600\%}$ of what it had been in $2001$ . Explain what $\text{600\%}$ means in this context.

Answers

1. $\frac{6}{100}$	3. $\frac{32}{1000}$	5. a) $\phantom{\rule{0.2em}{0ex}}\frac{57}{100}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}\text{57\%}$
7. a) $\phantom{\rule{0.2em}{0ex}}\frac{42}{100}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}\text{42\%}$	9. $\frac{1}{25}$	11. $\frac{17}{100}$
13. $\frac{13}{25}$	15. $\frac{5}{4}$	17. $\frac{3}{8}$
19. $\frac{23}{125}$	21. $\frac{19}{200}$	23. $\frac{4}{75}$
25. 0.05	27. 0.01	29. 0.63
31. 0.4	33. 1.15	35. 1.5
37. 0.214	39. 0.078	41. a) $\phantom{\rule{0.2em}{0ex}}\frac{3}{200}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.015$
43. a) $\phantom{\rule{0.2em}{0ex}}\frac{7023}{10,000}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.7023$	45. a) $\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.25$	47. a) $\phantom{\rule{0.2em}{0ex}}\frac{3}{5}\phantom{\rule{0.2em}{0ex}}$ b) $\phantom{\rule{0.2em}{0ex}}0.6$
49. 1%	51. 18%	53. 135%
55. 300%	57. 0.9%	59. 8.75%
61. 150%	63. 225.4%	65. 25%
67. 37.5%	69. 175%	71. 680%
73. 41.7%	75. $266.\stackrel{-}{6}\text{\%}$	77. 42.9%
79. 55.6%	81. 25%	83. 64%
87. $\frac{1}{4},\frac{1}{2},\frac{3}{4},1$ .	89. The Szetos sold their home for five times what they paid 30 years ago.

Attributions

This chapter has been adapted from “Understand Percent” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

3.3 Solve Proportions and their Applications

Learning Objectives

By the end of this section, you will be able to:

Use the definition of proportion
Solve proportions
Solve applications using proportions
Write percent equations as proportions
Translate and solve percent proportions

Use the Definition of Proportion

When two ratios or rates are equal, the equation relating them is called a proportion.

Proportion

A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d}$ , where $b\ne 0,d\ne 0$ .

The proportion states two ratios or rates are equal. The proportion is read $\text{``}a$ is to $b$ , as $c$ is to $d\text{''.}$

The equation $\frac{1}{2}=\frac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\frac{1}{2}=\frac{4}{8}$ is read $\text{``}1$ is to $2$ as $4$ is to $8\text{''.}$

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion $\frac{\text{20 students}}{\text{1 teacher}}=\frac{\text{60 students}}{\text{3 teachers}}$ we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

EXAMPLE 1

Write each sentence as a proportion:

$\phantom{\rule{0.2em}{0ex}}3$ is to $7$ as $15$ is to $35$ .
$\phantom{\rule{0.2em}{0ex}}5$ hits in $8$ at bats is the same as $30$ hits in $48$ at-bats.
$\phantom{\rule{0.2em}{0ex}}\text{\$1.50}$ for $6$ ounces is equivalent to $\text{\$2.25}$ for $9$ ounces.

Solution

a)
	3 is to 7 as 15 is to 35.
Write as a proportion.	$\frac{3}{7}=\frac{15}{35}$

b)
	5 hits in 8 at-bats is the same as 30 hits in 48 at-bats.
Write each fraction to compare hits to at-bats.	$\frac{\text{hits}}{\text{at-bats}}=\frac{\text{hits}}{\text{at-bats}}$
Write as a proportion.	$\frac{5}{8}=\frac{30}{48}$

c)
	$1.50 for 6 ounces is equivalent to $2.25 for 9 ounces.
Write each fraction to compare dollars to ounces.	$\frac{?}{\text{ounces}}=\frac{?}{\text{ounces}}$
Write as a proportion.	$\frac{1.50}{6}=\frac{2.25}{9}$

TRY IT 1.1

Write each sentence as a proportion:

$\phantom{\rule{0.2em}{0ex}}5$ is to $9$ as $20$ is to $36$ .
$\phantom{\rule{0.2em}{0ex}}7$ hits in $11$ at-bats is the same as $28$ hits in $44$ at-bats.
$\phantom{\rule{0.2em}{0ex}}\text{\$2.50}$ for $8$ ounces is equivalent to $\text{\$3.75}$ for $12$ ounces.

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{5}{9}=\frac{20}{36}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{7}{11}=\frac{28}{44}\phantom{\rule{0.2em}{0ex}}$
$\frac{2.50}{8}=\frac{3.75}{12}\phantom{\rule{0.2em}{0ex}}$

TRY IT 1.2

Write each sentence as a proportion:

$\phantom{\rule{0.2em}{0ex}}6$ is to $7$ as $36$ is to $42$ .
$\phantom{\rule{0.2em}{0ex}}8$ adults for $36$ children is the same as $12$ adults for $54$ children.
$\phantom{\rule{0.2em}{0ex}}\text{\$3.75}$ for $6$ ounces is equivalent to $\text{\$2.50}$ for $4$ ounces.

Answer

$\phantom{\rule{0.2em}{0ex}}\frac{6}{7}=\frac{36}{42}$
$\phantom{\rule{0.2em}{0ex}}\frac{8}{36}=\frac{12}{54}$
$\frac{3.75}{6}=\frac{2.50}{4}$

Look at the proportions $\frac{1}{2}=\frac{4}{8}$ and $\frac{2}{3}=\frac{6}{9}$ . From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?

To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cross products of a proportion are equal.

Cross Products of a Proportion

For any proportion of the form $\frac{a}{b}=\frac{c}{d}$ , where $b\ne 0,d\ne 0$ , its cross products are equal.

No Alt Text

Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.

EXAMPLE 2

Determine whether each equation is a proportion:

$\phantom{\rule{0.2em}{0ex}}\frac{4}{9}=\frac{12}{28}$
$\frac{17.5}{37.5}=\frac{7}{15}$

Solution

To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

a)

Find the cross products.	$28\cdot 4=112\phantom{\rule{2em}{0ex}}9\cdot 12=108$

Since the cross products are not equal, $28\cdot4\ne 9\cdot12$ , the equation is not a proportion.

b)

Find the cross products.	$15\cdot 17.5=262.5\phantom{\rule{2em}{0ex}}37.5\cdot 7=262.5$

Since the cross products are equal, $15\cdot17.5=37.5\cdot7$ , the equation is a proportion.

TRY IT 2.1

Determine whether each equation is a proportion:

$\phantom{\rule{0.2em}{0ex}}\frac{7}{9}=\frac{54}{72}$
$\frac{24.5}{45.5}=\frac{7}{13}$

Answer

TRY IT 2.2

Determine whether each equation is a proportion:

$\phantom{\rule{0.2em}{0ex}}\frac{8}{9}=\frac{56}{73}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{28.5}{52.5}=\frac{8}{15}$

Answer

Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.

EXAMPLE 3

Solve: $\frac{x}{63}=\frac{4}{7}$ .

Solution


To isolate $x$ , multiply both sides by the LCD, 63.
Simplify.
Divide the common factors.
Check: To check our answer, we substitute into the original proportion.


Show common factors.
Simplify.

TRY IT 3.1

Solve the proportion: $\frac{n}{84}=\frac{11}{12}$ .

Answer

TRY IT 3.2

Solve the proportion: $\frac{y}{96}=\frac{13}{12}$ .

Answer

104

When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

EXAMPLE 4

Solve: $\frac{144}{a}=\frac{9}{4}$ .

Solution

Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.


Find the cross products and set them equal.
Simplify.
Divide both sides by 9.
Simplify.
Check your answer:

Substitute a = 64
Show common factors.
Simplify.

Another method to solve this would be to multiply both sides by the LCD, $4a$ . Try it and verify that you get the same solution.

TRY IT 4.1

Solve the proportion: $\frac{91}{b}=\frac{7}{5}$ .

Answer

TRY IT 4.2

Solve the proportion: $\frac{39}{c}=\frac{13}{8}$ .

Answer

EXAMPLE 5

Solve: $\frac{52}{91}=\frac{-4}{y}$ .

Solution

Find the cross products and set them equal.

Simplify.
Divide both sides by 52.
Simplify.
Check:

Substitute y = −7
Show common factors.
Simplify.

TRY IT 5.1

Solve the proportion: $\frac{84}{98}=\frac{-6}{x}$ .

Answer

−7

TRY IT 5.2

Solve the proportion: $\frac{-7}{y}=\frac{105}{135}$ .

Answer

−9

Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

EXAMPLE 6

When pediatricians prescribe acetaminophen to children, they prescribe $5$ millilitre s (ml) of acetaminophen for every $25$ pounds of the child’s weight. If Zoe weighs $80$ pounds, how many millilitre s of acetaminophen will her doctor prescribe?

Solution

Identify what you are asked to find.	How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.	Let $a=$ ml of acetaminophen.
Write a sentence that gives the information to find it.	If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds?
Translate into a proportion.
Substitute given values—be careful of the units.
Multiply both sides by 80.
Multiply and show common factors.
Simplify.
Check if the answer is reasonable.
Yes. Since 80 is about 3 times 25, the medicine should be about 3 times 5.
Write a complete sentence.	The pediatrician would prescribe 16 ml of acetaminophen to Zoe.

You could also solve this proportion by setting the cross products equal.

TRY IT 6.1

Pediatricians prescribe $5$ millilitre s (ml) of acetaminophen for every $25$ pounds of a child’s weight. How many millilitre s of acetaminophen will the doctor prescribe for Emilia, who weighs $60$ pounds?

Answer

12 ml

TRY IT 6.2

For every $1$ kilogram (kg) of a child’s weight, pediatricians prescribe $15$ milligrams (mg) of a fever reducer. If Isabella weighs $12$ kg, how many milligrams of the fever reducer will the pediatrician prescribe?

Answer

180 mg

EXAMPLE 7

One brand of microwave popcorn has $120$ calories per serving. A whole bag of this popcorn has $3.5$ servings. How many calories are in a whole bag of this microwave popcorn?

Solution

Identify what you are asked to find.	How many calories are in a whole bag of microwave popcorn?
Choose a variable to represent it.	Let $c=$ number of calories.
Write a sentence that gives the information to find it.	If there are 120 calories per serving, how many calories are in a whole bag with 3.5 servings?
Translate into a proportion.
Substitute given values.
Multiply both sides by 3.5.
Multiply.
Check if the answer is reasonable.
Yes. Since 3.5 is between 3 and 4, the total calories should be between 360 (3⋅120) and 480 (4⋅120).
Write a complete sentence.	The whole bag of microwave popcorn has 420 calories.

TRY IT 7.1

Marissa loves the Caramel Macchiato at the coffee shop. The $16$ oz. medium size has $240$ calories. How many calories will she get if she drinks the large $20$ oz. size?

Answer

300

TRY IT 7.2

Yaneli loves Starburst candies, but wants to keep her snacks to $100$ calories. If the candies have $160$ calories for $8$ pieces, how many pieces can she have in her snack?

Answer

EXAMPLE 8

Josiah went to Mexico for spring break and changed $\text{\$325}$ dollars into Mexican pesos. At that time, the exchange rate had $\text{\$1}$ U.S. is equal to $12.54$ Mexican pesos. How many Mexican pesos did he get for his trip?

Solution

Identify what you are asked to find.	How many Mexican pesos did Josiah get?
Choose a variable to represent it.	Let $p=$ number of pesos.
Write a sentence that gives the information to find it.	If $1 U.S. is equal to 12.54 Mexican pesos, then $325 is how many pesos?
Translate into a proportion.
Substitute given values.
The variable is in the denominator, so find the cross products and set them equal.
Simplify.
Check if the answer is reasonable.
Yes, $100 would be $1,254 pesos. $325 is a little more than 3 times this amount.
Write a complete sentence.	Josiah has 4075.5 pesos for his spring break trip.

TRY IT 8.1

Yurianna is going to Europe and wants to change $\text{\$800}$ dollars into Euros. At the current exchange rate, $\text{\$1}$ Canadian dollar is equal to $0.65$ Euro. How many Euros will she have for her trip?

Answer

520 Euros

TRY IT 8.2

Corey and Nicole are traveling to Japan and need to exchange CAD $\text{\$600}$ into Japanese yen. If each dollar is $90.27$ yen, how many yen will they get?

Answer

54,162 yen

Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.

For example, $\text{60\%}=\frac{60}{100}$ and we can simplify $\frac{60}{100}=\frac{3}{5}$ . Since the equation $\frac{60}{100}=\frac{3}{5}$ shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:

$\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}$

$\phantom{\rule{1.2em}{0ex}}\frac{3}{5}=\frac{60}{100}$

Percent Proportion

The amount is to the base as the percent is to $100$ .

$\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}$

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

$\mathit{\text{The amount is to the base as the percent is to one hundred.}}$

We could also say:

$\mathit{\text{The amount out of the base is the same as the percent out of one hundred.}}$

First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

EXAMPLE 9

Translate to a proportion. What number is $\text{75\%}$ of $90?$

Solution

If you look for the word “of”, it may help you identify the base.

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=\text{number}$ .	$\frac{n}{90}=\frac{75}{100}$

TRY IT 9.1

Translate to a proportion: What number is $\text{60\%}$ of $105?$

Answer

$\frac{n}{105}=\frac{60}{100}$

TRY IT 9.2

Translate to a proportion: What number is $\text{40\%}$ of $85?$

Answer

$\frac{n}{85}=\frac{40}{100}$

EXAMPLE 10

Translate to a proportion. $19$ is $\text{25\%}$ of what number?

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=\text{number}$ .	$\frac{19}{n}=\frac{25}{100}$

TRY IT 10.1

Translate to a proportion: $36$ is $\text{25\%}$ of what number?

Answer

$\frac{36}{n}=\frac{25}{100}$

TRY IT 10.2

Translate to a proportion: $27$ is $\text{36\%}$ of what number?

Answer

$\frac{27}{n}=\frac{36}{100}$

EXAMPLE 11

Translate to a proportion. What percent of $27$ is $9?$

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $p=\text{percent}$ .	$\frac{9}{27}=\frac{p}{100}$

TRY IT 11.1

Translate to a proportion: What percent of $52$ is $39?$

Answer

$\frac{n}{100}=\frac{39}{52}$

TRY IT 11.2

Translate to a proportion: What percent of $92$ is $23?$

Answer

$\frac{n}{100}=\frac{23}{92}$

Translate and Solve Percent Proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.

EXAMPLE 12

Translate and solve using proportions: What number is $\text{45\%}$ of $80?$

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 100.
Simplify.
Check if the answer is reasonable.
Yes. 45 is a little less than half of 100 and 36 is a little less than half 80.
Write a complete sentence that answers the question.	36 is 45% of 80.

TRY IT 12.1

Translate and solve using proportions: What number is $\text{65\%}$ of $40?$

Answer

TRY IT 12.2

Translate and solve using proportions: What number is $\text{85\%}$ of $40?$

Answer

In the next example, the percent is more than $100$ , which is more than one whole. So the unknown number will be more than the base.

EXAMPLE 13

Translate and solve using proportions: $\text{125\%}$ of $25$ is what number?

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 100.
Simplify.
Check if the answer is reasonable.
Yes. 125 is more than 100 and 31.25 is more than 25.
Write a complete sentence that answers the question.	125% of 25 is 31.25.

TRY IT 13.1

Translate and solve using proportions: $\text{125\%}$ of $64$ is what number?

Answer

TRY IT 13.2

Translate and solve using proportions: $\text{175\%}$ of $84$ is what number?

Answer

147

Percents with decimals and money are also used in proportions.

EXAMPLE 14

Translate and solve: $\text{6.5\%}$ of what number is $\text{\$1.56}?$

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 6.5 to isolate the variable.
Simplify.
Check if the answer is reasonable.
Yes. 6.5% is a small amount and $1.56 is much less than $24.
Write a complete sentence that answers the question.	6.5% of $24 is $1.56.

TRY IT 14.1

Translate and solve using proportions: $\text{8.5\%}$ of what number is $\text{\$3.23}?$

Answer

TRY IT 14.2

Translate and solve using proportions: $\text{7.25\%}$ of what number is $\text{\$4.64}?$

Answer

EXAMPLE 15

Translate and solve using proportions: What percent of $72$ is $9?$

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 72.
Simplify.
Check if the answer is reasonable.
Yes. 9 is $\frac{1}{8}$ of 72 and $\frac{1}{8}$ is 12.5%.
Write a complete sentence that answers the question.	12.5% of 72 is 9.

TRY IT 15.1

Translate and solve using proportions: What percent of $72$ is $27?$

Answer

37.5%

TRY IT 15.2

Translate and solve using proportions: What percent of $92$ is $23?$

Answer

25%

Key Concepts

Proportion
- A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d}$ , where $b\ne 0$ , $d\ne 0$ .The proportion states two ratios or rates are equal. The proportion is read “ $a$ is to $b$ , as $c$ is to $d$ ”.
Cross Products of a Proportion
- For any proportion of the form $\frac{a}{b}=\frac{c}{d}$ , where $b\ne 0$ , its cross products are equal: $a\cdot d=b\cdot c$ .
Percent Proportion
- The amount is to the base as the percent is to 100. $\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}$

Glossary

proportion: A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d}$ , where $b\ne 0$ , $d\ne 0$ .The proportion states two ratios or rates are equal. The proportion is read “ $a$ is to $b$ , as $c$ is to $d$ ”.

Practice Makes Perfect

Use the Definition of Proportion

In the following exercises, write each sentence as a proportion.

1. $4$ is to $15$ as $36$ is to $135$ .	2. $7$ is to $9$ as $35$ is to $45$ .
3. $12$ is to $5$ as $96$ is to $40$ .	4. $15$ is to $8$ as $75$ is to $40$ .
5. $5$ wins in $7$ games is the same as $115$ wins in $161$ games.	6. $4$ wins in $9$ games is the same as $36$ wins in $81$ games.
7. $8$ campers to $1$ counsellor is the same as $48$ campers to $6$ counsellors.	8. $6$ campers to $1$ counselor is the same as $48$ campers to $8$ counselors.
9. $\text{\$9.36}$ for $18$ ounces is the same as $\text{\$2.60}$ for $5$ ounces.	10. $\text{\$3.92}$ for $8$ ounces is the same as $\text{\$1.47}$ for $3$ ounces.
11. $\text{\$18.04}$ for $11$ pounds is the same as $\text{\$4.92}$ for $3$ pounds.	12. $\text{\$12.42}$ for $27$ pounds is the same as $\text{\$5.52}$ for $12$ pounds.

In the following exercises, determine whether each equation is a proportion.

13. $\frac{7}{15}=\frac{56}{120}$	14. $\frac{5}{12}=\frac{45}{108}$
15. $\frac{11}{6}=\frac{21}{16}$	16. $\frac{9}{4}=\frac{39}{34}$
17. $\frac{12}{18}=\frac{4.99}{7.56}$	18. $\frac{9}{16}=\frac{2.16}{3.89}$
19. $\frac{13.5}{8.5}=\frac{31.05}{19.55}$	20. $\frac{10.1}{8.4}=\frac{3.03}{2.52}$

Solve Proportions

In the following exercises, solve each proportion.

21. $\frac{x}{56}=\frac{7}{8}$	22. $\frac{n}{91}=\frac{8}{13}$
23. $\frac{49}{63}=\frac{z}{9}$	24. $\frac{56}{72}=\frac{y}{9}$
25. $\frac{5}{a}=\frac{65}{117}$	26. $\frac{4}{b}=\frac{64}{144}$
27. $\frac{98}{154}=\frac{-7}{p}$	28. $\frac{72}{156}=\frac{-6}{q}$
29. $\frac{a}{-8}=\frac{-42}{48}$	30. $\frac{b}{-7}=\frac{-30}{42}$
31. $\frac{2.6}{3.9}=\frac{c}{3}$	32. $\frac{2.7}{3.6}=\frac{d}{4}$
33. $\frac{2.7}{j}=\frac{0.9}{0.2}$	34. $\frac{2.8}{k}=\frac{2.1}{1.5}$
35. $\frac{\frac{1}{2}}{1}=\frac{m}{8}$	36. $\frac{\frac{1}{3}}{3}=\frac{9}{n}$

Solve Applications Using Proportions

In the following exercises, solve the proportion problem.

37. Pediatricians prescribe $5$ millilitre s (ml) of acetaminophen for every $25$ pounds of a child’s weight. How many millilitres of acetaminophen will the doctor prescribe for Jocelyn, who weighs $45$ pounds?	38. Brianna, who weighs $6$ kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at $15$ milligrams (mg) for every $1$ kilogram (kg) of the child’s weight. How many milligrams will the doctor prescribe?
39. At the gym, Carol takes her pulse for $10$ sec and counts $19$ beats. How many beats per minute is this? Has Carol met her target heart rate of $140$ beats per minute?	40. Kevin wants to keep his heart rate at $160$ beats per minute while training. During his workout he counts $27$ beats in $10$ seconds. How many beats per minute is this? Has Kevin met his target heart rate?
41. A new energy drink advertises $106$ calories for $8$ ounces. How many calories are in $12$ ounces of the drink?	42. One $12$ ounce can of soda has $150$ calories. If Josiah drinks the big $32$ ounce size from the local mini-mart, how many calories does he get?
43. Karen eats $\frac{1}{2}$ cup of oatmeal that counts for $2$ points on her weight loss program. Her husband, Joe, can have $3$ points of oatmeal for breakfast. How much oatmeal can he have?	44. An oatmeal cookie recipe calls for $\frac{1}{2}$ cup of butter to make $4$ dozen cookies. Hilda needs to make $10$ dozen cookies for the bake sale. How many cups of butter will she need?
45. Janice is traveling to the US and will change $\text{\$250}$ Canadian dollars into US dollars. At the current exchange rate, $\text{\$1}$ Canadian is equal to $\text{\$0.70}$ US. How many Canadian dollars will she get for her trip?	46. Todd is traveling to Mexico and needs to exchange $\text{\$450}$ into Mexican pesos. If each dollar is worth $17.20$ pesos, how many pesos will he get for his trip?
47. Steve changed $\text{\$782}$ into $507.08$ Euros. How many Euros did he receive per Canadian dollar?	48. Martha changed $\text{\$350}$ Canadian into $392.28$ Australian dollars. How many Australian dollars did she receive per US dollar?
49. At the laundromat, Lucy changed $\text{\$12.00}$ into quarters. How many quarters did she get?	50. When she arrived at a casino, Gerty changed $\text{\$20}$ into nickels. How many nickels did she get?
51. Jesse’s car gets $30$ miles per gallon of gas. If Toronto is $285$ miles away, how many gallons of gas are needed to get there and then home? If gas is $\text{\$3.09}$ per gallon, what is the total cost of the gas for the trip?	52. Danny wants to drive to Banff to see his grandfather. Banff is $370$ miles from Danny’s home and his car gets $18.5$ miles per gallon. How many gallons of gas will Danny need to get to and from Banff? If gas is $\text{\$3.19}$ per gallon, what is the total cost for the gas to drive to see his grandfather?
53. Hugh leaves early one morning to drive from his home in White Rock to go to Edmonton, $702$ miles away. After $3$ hours, he has gone $190$ miles. At that rate, how long will the whole drive take?	54. Kelly leaves her home in Seattle to drive to Spokane, a distance of $280$ miles. After $2$ hours, she has gone $152$ miles. At that rate, how long will the whole drive take?
55. Phil wants to fertilize his lawn. Each bag of fertilizer covers about $4,000$ square feet of lawn. Phil’s lawn is approximately $13,500$ square feet. How many bags of fertilizer will he have to buy?	56. April wants to paint the exterior of her house. One gallon of paint covers about $350$ square feet, and the exterior of the house measures approximately $2000$ square feet. How many gallons of paint will she have to buy?

Write Percent Equations as Proportions

In the following exercises, translate to a proportion.

57. What number is $\text{35\%}$ of $250?$	58. What number is $\text{75\%}$ of $920?$
59. What number is $\text{110\%}$ of $47?$	60. What number is $\text{150\%}$ of $64?$
61. $45$ is $\text{30\%}$ of what number?	62. $25$ is $\text{80\%}$ of what number?
63. $90$ is $\text{150\%}$ of what number?	64. $77$ is $\text{110\%}$ of what number?
64. $77$ is $\text{110\%}$ of what number?	65. What percent of $85$ is $17?$
66. What percent of $92$ is $46?$	67. What percent of $260$ is $340?$
68. What percent of $180$ is $220?$

Translate and Solve Percent Proportions

In the following exercises, translate and solve using proportions.

69. What number is $\text{65\%}$ of $180?$	70. What number is $\text{55\%}$ of $300?$
71. $\text{18\%}$ of $92$ is what number?	72. $\text{22\%}$ of $74$ is what number?
73. $\text{175\%}$ of $26$ is what number?	74. $\text{250\%}$ of $61$ is what number?
75. What is $\text{300\%}$ of $488?$	76. What is $\text{500\%}$ of $315?$
77. $\text{17\%}$ of what number is $\text{\$7.65}?$	78. $\text{19\%}$ of what number is $\text{\$6.46}?$
79. $\text{\$13.53}$ is $\text{8.25\%}$ of what number?	80. $\text{\$18.12}$ is $\text{7.55\%}$ of what number?
81. What percent of $56$ is $14?$	82. What percent of $80$ is $28?$
83. What percent of $96$ is $12?$	84. What percent of $120$ is $27?$

Everyday Math

85. Mixing a concentrate Sam bought a large bottle of concentrated cleaning solution at the warehouse store. He must mix the concentrate with water to make a solution for washing his windows. The directions tell him to mix $3$ ounces of concentrate with $5$ ounces of water. If he puts $12$ ounces of concentrate in a bucket, how many ounces of water should he add? How many ounces of the solution will he have altogether?

86. Mixing a concentrate Travis is going to wash his car. The directions on the bottle of car wash concentrate say to mix $2$ ounces of concentrate with $15$ ounces of water. If Travis puts $6$ ounces of concentrate in a bucket, how much water must he mix with the concentrate?

Writing Exercises

87. To solve “what number is $\text{45\%}$ of $350\text{''}$ do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

88. To solve “what percent of $125$ is $25\text{''}$ do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

Answers

1. $\frac{4}{15}=\frac{36}{135}$	3. $\frac{12}{5}=\frac{96}{40}$	5. $\frac{5}{7}=\frac{115}{161}$
7. $\frac{8}{1}=\frac{48}{6}$	9. $\frac{9.36}{18}=\frac{2.60}{5}$	11. $\frac{18.04}{11}=\frac{4.92}{3}$
13. yes	15. no	17. no
19. yes	21. 49	23. 47
25. 9	27. -11	29. 7
31. 2	33. 0.6	35. 4
37. 9 ml	39. 114, no	41. 159 cal
43. $\frac{3}{4}\phantom{\rule{0.3em}{0ex}}\text{cup}$	45. $175.00	47. 0.65
49. 48 quarters	51. 19, $58.71	53. 11.1 hours
55. 4 bags	57. $\frac{n}{250}=\frac{35}{100}$	59. $\frac{n}{47}=\frac{110}{100}$
61. $\frac{45}{n}=\frac{30}{100}$	63. $\frac{90}{n}=\frac{150}{100}$	65. $\frac{17}{85}=\frac{p}{100}$
67. $\frac{340}{260}=\frac{p}{100}$	69. 117	70. 165
71. 16.56	73. 45.5	75. 1464
77. $45	79. $164	81. 25%
83. 12.5%	85. 20, 32	87. Answers will vary.

Attributions

This chapter has been adapted from “Solve Proportions and their Applications” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

3.4 Solve General Applications of Percent

Learning Objectives

By the end of this section, you will be able to:

Translate and solve basic percent equations
Solve applications of percent
Find percent increase and percent decrease

Translate and Solve Basic Percent Equations

In the last section, we solved percent problems by setting them up as proportions. That is the best method available when you did not have the tools of algebra. Now, in this section we will translate word sentences into algebraic equations, and then solve the percent equations.

We’ll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

When Kim and her friends went on a road trip to Vancouver, they ate lunch at Marta’s Cafe Tower. The bill came to $\text{\$80}$ . They wanted to leave a $\text{20\%}$ tip. What amount would the tip be?

To solve this, we want to find what amount is $\text{20\%}$ of $\text{\$80}$ . The $\text{\$80}$ is called the base. The amount of the tip would be $0.20\left(80\right)$ , or $\text{\$16}$ See (Figure 1). To find the amount of the tip, we multiplied the percent by the base.

A $\text{20\%}$ tip for an $\text{\$80}$ restaurant bill comes out to $\text{\$16}$ .

Figure 1.(credit: Marta Oraniewicz)

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.

EXAMPLE 1

What number is $\text{35\%}$ of $90?$

Solution

Translate into algebra. Let $n=\phantom{\rule{0.2em}{0ex}}$ the number.
Multiply.
	$31.5$ is $35\%$ of $90$

TRY IT 1.1

What number is $\text{45\%}$ of $80?$

Answer

TRY IT 1.2

What number is $\text{55\%}$ of $60?$

Answer

EXAMPLE 2

$\text{125\%}$ of $28$ is what number?

Solution

Translate into algebra. Let $a\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}$ the number.
Multiply.
	$125\%$ of $28$ is $35$ .

Remember that a percent over $100$ is a number greater than $1$ . We found that $\text{125\%}$ of $28$ is $35$ , which is greater than $28$ .

TRY IT 2.1

$\text{150\%}$ of $78$ is what number?

Answer

117

TRY IT 2.2

$\text{175\%}$ of $72$ is what number?

Answer

126

In the next examples, we are asked to find the base.

EXAMPLE 3

Translate and solve: $36$ is $\text{75\%}$ of what number?

Solution

Translate. Let $b=$ the number.
Divide both sides by 0.75.
Simplify.

TRY IT 3.1

$17$ is $\text{25\%}$ of what number?

Answer

TRY IT 3.2

$40$ is $\text{62.5\%}$ of what number?

Answer

EXAMPLE 4

$\text{6.5\%}$ of what number is $\text{\$1.17}?$

Solution

Translate. Let $b=$ the number.
Divide both sides by 0.065.
Simplify.

TRY IT 4.1

$\text{7.5\%}$ of what number is $\text{\$1.95}?$

Answer

$26

TRY IT 4.1

$\text{8.5\%}$ of what number is $\text{\$3.06}?$

Answer

$36

In the next examples, we will solve for the percent.

EXAMPLE 5

What percent of $36$ is $9?$

Solution

Translate into algebra. Let $p=$ the percent.
Divide by 36.
Simplify.
Convert to decimal form.
Convert to percent.

TRY IT 5.1

What percent of $76$ is $57?$

Answer

75%

TRY IT 5.2

What percent of $120$ is $96?$

Answer

80%

EXAMPLE 6

$144$ is what percent of $96?$

Solution

Translate into algebra. Let $p=$ the percent.
Divide by 96.
Simplify.
Convert to percent.

TRY IT 6.1

$110$ is what percent of $88?$

Answer

125%

TRY IT 6.2

$126$ is what percent of $72?$

Answer

175%

Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we’ll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

We will update the strategy we used in our earlier applications to include equations now. Notice that we will translate a sentence into an equation.

HOW TO: Solve an Application

Identify what you are asked to find and choose a variable to represent it.
Write a sentence that gives the information to find it.
Translate the sentence into an equation.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Write a complete sentence that answers the question.

Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we’ll solve involve everyday situations, you can rely on your own experience.

EXAMPLE 7

Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was $\text{\$68.50}$ . They want to leave an $\text{18\%}$ tip. If the tip will be $\text{18\%}$ of the total bill, how much should the tip be?

Solution

What are you asked to find?	The amount of the tip
Choose a variable to represent it.	Let $t=$ amount of tip.
Write a sentence that give the information to find it.	The tip is 18% of the total bill.
Translate the sentence into an equation.
Multiply.
Check. Is this answer reasonable?
If we approximate the bill to $70 and the percent to 20%, we would have a tip of $14. So a tip of $12.33 seems reasonable.
Write a complete sentence that answers the question.	The couple should leave a tip of $12.33.

TRY IT 7.1

Cierra and her sister enjoyed a special dinner in a restaurant, and the bill was $\text{\$81.50}$ . If she wants to leave $\text{18\%}$ of the total bill as her tip, how much should she leave?

Answer

$14.67

TRY IT 7.2

Kimngoc had lunch at her favorite restaurant. She wants to leave $\text{15\%}$ of the total bill as her tip. If her bill was $\text{\$14.40}$ , how much will she leave for the tip?

Answer

$2.16

EXAMPLE 8

The label on Masao’s breakfast cereal said that one serving of cereal provides $85$ milligrams (mg) of potassium, which is $\text{2\%}$ of the recommended daily amount. What is the total recommended daily amount of potassium?

The figures shows the nutrition facts for cereal.

Solution

What are you asked to find?	the total amount of potassium recommended
Choose a variable to represent it.	Let $a=$ total amount of potassium.
Write a sentence that gives the information to find it.	85 mg is 2% of the total amount.
Translate the sentence into an equation.
Divide both sides by 0.02.
Simplify.
Check: Is this answer reasonable?
Yes. 2% is a small percent and 85 is a small part of 4,250.
Write a complete sentence that answers the question.	The amount of potassium that is recommended is 4250 mg.

TRY IT 8.1

One serving of wheat square cereal has $7$ grams of fiber, which is $\text{29\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?

Answer

24.1 grams

TRY IT 8.2

One serving of rice cereal has $190$ mg of sodium, which is $\text{8\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?

Answer

2,375 mg

EXAMPLE 9

Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was $480$ calories, and had $240$ calories of fat. What percent of the total calories in each brownie comes from fat?

Solution

What are you asked to find?	the percent of the total calories from fat
Choose a variable to represent it.	Let $p=$ percent from fat.
Write a sentence that gives the information to find it.	What percent of 480 is 240?
Translate the sentence into an equation.
Divide both sides by 480.
Simplify.
Convert to percent form.
Check. Is this answer reasonable?
Yes. 240 is half of 480, so 50% makes sense.
Write a complete sentence that answers the question.	Of the total calories in each brownie, 50% is fat.

TRY IT 9.1

Veronica is planning to make muffins from a mix. The package says each muffin will be $230$ calories and $60$ calories will be from fat. What percent of the total calories is from fat? (Round to the nearest whole percent.)

Answer

26%

Exercises

The brownie mix Ricardo plans to use says that each brownie will be $190$ calories, and $70$ calories are from fat. What percent of the total calories are from fat?

Answer

37%

Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.

To find the percent increase, first we find the amount of increase, which is the difference between the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.

HOW TO: Find Percent Increase

Step 1. Find the amount of increase.

$\text{increase}=\text{new amount}-\text{original amount}$

Step 2. Find the percent increase as a percent of the original amount.

EXAMPLE 10

In $2017$ , university tuition fees in Canada for domestic students increased from $\text{\$26}$ per school year to $\text{\$36}$ per school year. Find the percent increase. (Round to the nearest tenth of a percent.)

Solution

What are you asked to find?	the percent increase
Choose a variable to represent it.	Let $p=$ percent.
Find the amount of increase.
Find the percent increase.	The increase is what percent of the original amount?
Translate to an equation.
Divide both sides by 26.
Round to the nearest thousandth.
Convert to percent form.
Write a complete sentence.	The new fees represent a 38.4\% increase over the old fees.

TRY IT 10.1

In $2011$ , the IRS increased the deductible mileage cost to $55.5$ cents from $51$ cents. Find the percent increase. (Round to the nearest tenth of a percent.)

Answer

8.8%

TRY IT 10.2

In $1984$ , the standard bus fare in Vancouver was $\text{\$1.25}$ . In $2008$ , the standard bus fare was $\text{\$2.50}$ . Find the percent increase. (Round to the nearest tenth of a percent.)

Answer

50%

Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference between the original amount and the final amount. Then we find what percent the amount of decrease is of the original amount.

HOW TO: Find Percent Decrease

Find the amount of decrease.
- $\text{decrease}=\text{original amount}-\text{new amount}$
Find the percent decrease as a percent of the original amount.

EXAMPLE 11

The average price of a gallon of gas in one city in June $2014$ was $\text{\$3.71}$ . The average price in that city in July was $\text{\$3.64}$ . Find the percent decrease.

Solution

What are you asked to find?	the percent decrease
Choose a variable to represent it.	Let $p=$ percent.
Find the amount of decrease.
Find the percent of decrease.	The decrease is what percent of the original amount?
Translate to an equation.
Divide both sides by 3.71.
Round to the nearest thousandth.
Convert to percent form.
Write a complete sentence.	The price of gas decreased 1.9%.

TRY IT 11.1

The population of one city was about $672,000$ in $2010$ . The population of the city is projected to be about $630,000$ in $2020$ . Find the percent decrease. (Round to the nearest tenth of a percent.)

Answer

6.3%

TRY IT 11.2

Last year Sheila’s salary was $\text{\$42,000}$ . Because of furlough days, this year her salary was $\text{\$37,800}$ . Find the percent decrease. (Round to the nearest tenth of a percent.)

Answer

10%

Access Additional Online Resources

Percent Increase and Percent Decrease Visualization

Key Concepts

Solve an application.
1. Identify what you are asked to find and choose a variable to represent it.
2. Write a sentence that gives the information to find it.
3. Translate the sentence into an equation.
4. Solve the equation using good algebra techniques.
5. Write a complete sentence that answers the question.
6. Check the answer in the problem and make sure it makes sense.
Find percent increase.
1. Find the amount of increase:
  $\text{increase}=\text{new amount}-\text{original amount}$
2. Find the percent increase as a percent of the original amount.
Find percent decrease.
1. Find the amount of decrease.
  $\text{decrease}=\text{original amount}-\text{new amount}$
2. Find the percent decrease as a percent of the original amount.

Glossary

percent increase: The percent increase is the percent the amount of increase is of the original amount.

percent decrease: The percent decrease is the percent the amount of decrease is of the original amount.

Practice Makes Perfect

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

1. What number is $\text{45\%}$ of $120?$	2. What number is $\text{65\%}$ of $100?$
3. What number is $\text{24\%}$ of $112?$	4. What number is $\text{36\%}$ of $124?$
5. $\text{250\%}$ of $65$ is what number?	6. $\text{150\%}$ of $90$ is what number?
7. $\text{800\%}$ of $2,250$ is what number?	8. $\text{600\%}$ of $1,740$ is what number?
9. $28$ is $\text{25\%}$ of what number?	10. $36$ is $\text{25\%}$ of what number?
11. $81$ is $\text{75\%}$ of what number?	12. $93$ is $\text{75\%}$ of what number?
13. $\text{8.2\%}$ of what number is $\text{\$2.87}?$	14. $\text{6.4\%}$ of what number is $\text{\$2.88}?$
15. $\text{11.5\%}$ of what number is $\text{\$108.10}?$	16. $\text{12.3\%}$ of what number is $\text{\$92.25}?$
17. What percent of $260$ is $78?$	18. What percent of $215$ is $86?$
19. What percent of $1,500$ is $540?$	20. What percent of $1,800$ is $846?$
21. $30$ is what percent of $20?$	22. $50$ is what percent of $40?$
23. $840$ is what percent of $480?$	24. $790$ is what percent of $395?$

Solve Applications of Percents

In the following exercises, solve the applications of percents.

25. Geneva treated her parents to dinner at their favorite restaurant. The bill was $\text{\$74.25}$ . She wants to leave $\text{16\%}$ of the total bill as a tip. How much should the tip be?	26. When Hiro and his co-workers had lunch at a restaurant the bill was $\text{\$90.50}$ . They want to leave $\text{18\%}$ of the total bill as a tip. How much should the tip be?
27. Trong has $\text{12\%}$ of each paycheck automatically deposited to his savings account. His last paycheck was $\text{\$2,165}$ . How much money was deposited to Trong’s savings account?	28. Cherise deposits $\text{8\%}$ of each paycheck into her retirement account. Her last paycheck was $\text{\$1,485}$ . How much did Cherise deposit into her retirement account?
29. One serving of oatmeal has $8$ grams of fiber, which is $\text{33\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?	30. One serving of trail mix has $67$ grams of carbohydrates, which is $\text{22\%}$ of the recommended daily amount. What is the total recommended daily amount of carbohydrates?
31. A bacon cheeseburger at a popular fast food restaurant contains $2,070$ milligrams (mg) of sodium, which is $\text{86\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?	32. A grilled chicken salad at a popular fast food restaurant contains $650$ milligrams (mg) of sodium, which is $\text{27\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?
33. The nutrition fact sheet at a fast food restaurant says the fish sandwich has $380$ calories, and $171$ calories are from fat. What percent of the total calories is from fat?	34. The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has $190$ calories, and $114$ calories are from fat. What percent of the total calories is from fat?
35. Emma gets paid $\text{\$3,000}$ per month. She pays $\text{\$750}$ a month for rent. What percent of her monthly pay goes to rent?	36. Dimple gets paid $\text{\$3,200}$ per month. She pays $\text{\$960}$ a month for rent. What percent of her monthly pay goes to rent?

Find Percent Increase and Percent Decrease

In the following exercises, find the percent increase or percent decrease.

37. Tamanika got a raise in her hourly pay, from $\text{\$15.50}$ to $\text{\$17.55}$ . Find the percent increase.	38. Ayodele got a raise in her hourly pay, from $\text{\$24.50}$ to $\text{\$25.48}$ . Find the percent increase.
39. According to Statistics Canada, annual international graduate student fees in Canada rose from about $\text{\$13,000}$ in $2015$ to about $\text{\$15,000}$ in $2019$ . Find the percent increase.	40. The price of a share of one stock rose from $\text{\$12.50}$ to $\text{\$50}$ . Find the percent increase.
41. According to Time magazine $\left(\text{7/19/2011}\right)$ annual global seafood consumption rose from $22$ pounds per person in $1960$ to $38$ pounds per person today. Find the percent increase. (Round to the nearest tenth of a percent.)	42. In one month, the median home price in the Northeast rose from $\text{\$225,400}$ to $\text{\$241,500}$ . Find the percent increase. (Round to the nearest tenth of a percent.)
43. A grocery store reduced the price of a loaf of bread from $\text{\$2.80}$ to $\text{\$2.73}$ . Find the percent decrease.	44. The price of a share of one stock fell from $\text{\$8.75}$ to $\text{\$8.54}$ . Find the percent decrease.
45. Hernando’s salary was $\text{\$49,500}$ last year. This year his salary was cut to $\text{\$44,055}$ . Find the percent decrease.	46. From $2000$ to $2010$ , the population of Detroit fell from about $951,000$ to about $714,000$ . Find the percent decrease. (Round to the nearest tenth of a percent.)
47. In one month, the median home price in the West fell from $\text{\$203,400}$ to $\text{\$192,300}$ . Find the percent decrease. (Round to the nearest tenth of a percent.)	48. Sales of video games and consoles fell from $\text{\$1,150}$ million to $\text{\$1,030}$ million in one year. Find the percent decrease. (Round to the nearest tenth of a percent.)

Everyday Math

49. Tipping At the campus coffee cart, a medium coffee costs $\text{\$1.65}$ . MaryAnne brings $\text{\$2.00}$ with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?

50. Late Fees Alison was late paying her credit card bill of $\text{\$249}$ . She was charged a $\text{5\%}$ late fee. What was the amount of the late fee?

Writing Exercises

51. Without solving the problem $``44$ is $\text{80\%}$ of what number”, think about what the solution might be. Should it be a number that is greater than $44$ or less than $44?$ Explain your reasoning.	52. Without solving the problem “What is $\text{20\%}$ of $300?''$ think about what the solution might be. Should it be a number that is greater than $300$ or less than $300?$ Explain your reasoning.
53. After returning from vacation, Alex said he should have packed $\text{50\%}$ fewer shorts and $\text{200\%}$ more shirts. Explain what Alex meant.	54. Because of road construction in one city, commuters were advised to plan their Monday morning commute to take $\text{150\%}$ of their usual commuting time. Explain what this means.

Answers

1. 54	3. 26.88	5. 162.5
7. 18,000	9. 112	11. 108
13. $35	15. $940	17. 30%
19. 36%	21. 150%	23. 175%
25. $11.88	27. $259.80	29. 24.2 grams
31. 2,407 grams	33. 45%	35. 25%
37. 13.2%	39. 15%	41. 72.7%
43. 2.5%	45. 11%	47. 5.5%
49. 21.2%	51. The original number should be greater than 44.80% is less than 100%, so when 80% is converted to a decimal and multiplied to the base in the percent equation, the resulting amount of 44 is less. 44 is only the larger number in cases where the percent is greater than 100%.	53. Alex should have packed half as many shorts and twice as many shirts.

Attributions

This chapter has been adapted from “Solve General Applications of Percent” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

3.5 Chapter Review

Review Exercises

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction. Simplify the answer if possible.

1. $56$ to $32$	2. $28$ to $40$
3. $1.2$ to $1.8$	4. $3.5$ to $0.5$
5. $2\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}5\frac{1}{4}$	6. $1\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}1\frac{5}{8}$
7. $28$ inches to $3$ feet	8. $64$ ounces to $30$ ounces

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction. Simplify the answer if possible.

9. $90$ pounds per $7.5$ square inches	10. $180$ calories per $8$ ounces
11. $\text{\$612.50}$ for $35$ hours	12. $126$ miles in $4$ hours

Find Unit Rates

In the following exercises, find the unit rate.

13. $90$ pounds per $7.5$ square inches	14. $180$ calories per $8$ ounces
15. $\text{\$612.50}$ for $35$ hours	16. $126$ miles in $4$ hours

Find Unit Price

In the following exercises, find the unit price.

17.Highlighters: $6$ for $\text{\$2.52}$	18. T-shirts: $3$ for $\text{\$8.97}$
19. Anna bought a pack of $8$ kitchen towels for $\text{\$13.20}$ . How much did each towel cost? Round to the nearest cent if necessary.	20. An office supply store sells a box of pens for $\text{\$11}$ . The box contains $12$ pens. How much does each pen cost?

In the following exercises, find each unit price and then determine the better buy.

21.Vitamins: $60$ tablets for $\text{\$6.49}$ or $100$ for $\text{\$11.99}?$

22. Shampoo: $12$ ounces for $\text{\$4.29}$ or $22$ ounces for $\text{\$7.29}?$

Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.

23. $a$ adults to $45$ children	24. $535$ miles per $h\phantom{\rule{0.2em}{0ex}}\text{hours}$
25. the ratio of $19$ and the sum of $3$ and $n$	26. the ratio of $4y$ and the difference of $x$ and $10$

In the following exercises, write each percent as a ratio.

27. $\text{32\%}$ admission rate for the university

28. $\text{53.3\%}$ rate of college students with student loans

In the following exercises, write as a ratio and then as a percent.

29. $13$ out of $100$ architects are women.

30. $9$ out of every $100$ nurses are men.

In the following exercises, convert each percent to a fraction.

31. $\text{48\%}$	32. $\text{175\%}$
33. $\text{64.1\%}$	34. $8\frac{1}{4}\%$

In the following exercises, convert each percent to a decimal.

35. $\text{6\%}$	36. $\text{23\%}$
37. $\text{128\%}$	38. 4.9%

In the following exercises, convert each percent to a) a simplified fraction and b) a decimal.

39. In $2016,\text{17\%}$ of the Canadian population was age $65$ or over. (Source: www12.statcan.gc.ca)	40. In $2016,\text{16.6\%}$ of the Canadian population was under $15$ years old. (Source: www12.statcan.gc.ca)
41. When a die is tossed, the probability it will land with an even number of dots on the top side is $\text{50\%}$ .	42. A couple plans to have three children. The probability they will all be girls is $\text{12.5\%}$ .

In the following exercises, convert each decimal to a percent.

43. $0.04$	44. $0.15$
45. $2.82$	46. $3$
47. $0.003$	48. $1.395$

In the following exercises, convert each fraction to a percent.

49. $\frac{3}{4}$	50. $\frac{11}{5}$
51. $3\frac{5}{8}$	52. $\frac{2}{9}$
53. According to the Centers for Disease Control, $\frac{2}{5}$ of adults do not take a vitamin or supplement.	54. According to the Centers for Disease Control, among adults who do take a vitamin or supplement, $\frac{3}{4}$ take a multivitamin.

In the following exercises, translate and solve.

55. What number is $\text{46\%}$ of $350?$	56. $\text{120\%}$ of $55$ is what number?
57. $84$ is $\text{35\%}$ of what number?	58. $15$ is $\text{8\%}$ of what number?
59. $\text{200\%}$ of what number is $50?$	60. $\text{7.9\%}$ of what number is $\text{\$4.74}?$
61. What percent of $120$ is $81.6?$	62. What percent of $340$ is $595?$

Solve General Applications of Percents

In the following exercises, solve.

63. When Aurelio and his family ate dinner at a restaurant, the bill was $\text{\$83.50}$ . Aurelio wants to leave $\text{20\%}$ of the total bill as a tip. How much should the tip be?	64. One granola bar has $2$ grams of fiber, which is $\text{8\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?
65. The nutrition label on a package of granola bars says that each granola bar has $190$ calories, and $54$ calories are from fat. What percent of the total calories is from fat?	66. Elsa gets paid $\text{\$4,600}$ per month. Her car payment is $\text{\$253}$ . What percent of her monthly pay goes to her car payment?

67. Marta got a gift of $\text{\$1900}$ from her uncle. She spent $\text{35\%}$ of that money for her trip to Victoria. How much money she has left?.

68. Last year Bernard bought a new car for $\text{\$30,000}$ . If the value of the car depreciated $\text{20\%}$ every year , find the value of the car this year.

Solve Proportions and their Applications

In the following exercises, write each sentence as a proportion.

69. $3$ is to $8$ as $12$ is to $32$ .	70. $95$ miles to $3$ gallons is the same as $475$ miles to $15$ gallons.
71. $1$ teacher to $18$ students is the same as $23$ teachers to $414$ students.	72. $\text{\$7.35}$ for $15$ ounces is the same as $\text{\$2.94}$ for $6$ ounces.

In the following exercises, determine whether each equation is a proportion.

73. $\frac{5}{13}=\frac{30}{78}$	74. $\frac{16}{7}=\frac{48}{23}$
75. $\frac{12}{18}=\frac{6.99}{10.99}$	76. $\frac{11.6}{9.2}=\frac{37.12}{29.44}$

In the following exercises, solve each proportion.

77. $\frac{x}{36}=\frac{5}{9}$	78. $\frac{7}{a}=\frac{-6}{84}$
79. $\frac{1.2}{1.8}=\frac{d}{6}$	80. $\frac{\frac{1}{2}}{2}=\frac{m}{20}$

In the following exercises, solve the proportion problem.

81. The children’s dosage of acetaminophen is $5$ millilitre s (ml) for every $25$ pounds of a child’s weight. How many millilitre s of acetaminophen will be prescribed for a $60$ pound child?	82. After a workout, Dennis takes his pulse for $10$ sec and counts $21$ beats. How many beats per minute is this?
83. An $8$ ounce serving of ice cream has $272$ calories. If Lavonne eats $10$ ounces of ice cream, how many calories does she get?	84. Alma is going to Europe and wants to exchange $\text{\$1,200}$ into Euros. If each dollar is $0.65$ Euros, how many Euros will Alma get?
85. Zack wants to drive from Abbotsford to Banff, a distance of $494$ miles. If his car gets $38$ miles to the gallon, how many gallons of gas will Zack need to get to Banff?	86. Teresa is planning a party for $100$ people. Each gallon of punch will serve $18$ people. How many gallons of punch will she need?

In the following exercises, translate to a proportion.

87. What number is $\text{62\%}$ of $395?$	88. $42$ is $\text{70\%}$ of what number?
89. What percent of $1,000$ is $15?$	90. What percent of $140$ is $210?$

In the following exercises, translate and solve using proportions.

91. What number is $\text{85\%}$ of $900?$	92. $\text{6\%}$ of what number is $\text{\$24}?$
93. $\text{\$3.51}$ is $\text{4.5\%}$ of what number?	94. What percent of $3,100$ is $930?$

In the following exercises, convert each percent to a) a decimal b) a simplified fraction.

95. $\text{24\%}$	96. $\text{5\%}$
97. $\text{350\%}$

In the following exercises, convert each fraction to a percent. (Round to $3$ decimal places if needed.)

98. $\frac{7}{8}$	99. $\frac{1}{3}$
100. $\frac{11}{12}$

In the following exercises, solve the percent problem.

101. $65$ is what percent of $260?$	102. What number is $\text{27\%}$ of $3,000?$
103. $\text{150\%}$ of what number is $60?$	104. Write as a proportion: $4$ gallons to $144$ miles is the same as $10$ gallons to $360$ miles.
105. Vin read $10$ pages of a book in $12$ minutes. At that rate, how long will it take him to read $35$ pages?

Review Answers

1. $\frac{7}{4}$	3. $\frac{2}{3}$	5. $\frac{4}{9}$
7. $\frac{7}{9}$	9. $\frac{90\phantom{\rule{0.2em}{0ex}}\text{pounds}}{7.5\phantom{\rule{0.2em}{0ex}}\text{square inches}}$	11. $\frac{\text{\$612.50}}{35\phantom{\rule{0.2em}{0ex}}\text{hours}}$
13. 12 pounds/sq.in.	15. $17.50/hour	17. $0.42
19. $1.65	21. $0.11, $0.12; 60 tablets for $6.49	23. $\frac{a\phantom{\rule{0.2em}{0ex}}\text{adults}}{45\phantom{\rule{0.2em}{0ex}}\text{children}}$
25. $\frac{19}{3+n}$	27. $\frac{32}{100}$	29. $\frac{13}{100},\text{13\%}$
31. $\frac{12}{25}$	33. $\frac{641}{1000}$	35. 0.06
37. 1.28	39. a) $\phantom{\rule{0.2em}{0ex}}\frac{17}{100}$ b) $\phantom{\rule{0.2em}{0ex}}0.17$	41. a) $\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$ b) $\phantom{\rule{0.2em}{0ex}}0.5$
43. 4%	45. 282%	47. 0.3%
49. 75%	51. 362.5%	53. 40%
55. 161	57. 240	59. 25
61. 68%	63. $16.70	65. 28.4%
67. 1235	69. $\frac{3}{8}=\frac{12}{32}$	71. $\frac{1}{18}=\frac{23}{414}$
73. yes	75. no	77. 20
79. 4	81. 12	83. 340
87. $\frac{x}{395}=\frac{62}{100}$	89. $\frac{x}{100}=\frac{15}{1000}$	91.765
93. $78	95. $0.24,\frac{6}{25}$	97. $3.5,3\frac{1}{2}$
99. 33.333%	101.25%	103. 40
105. 42

Chapter Test

1. Write a ratio as a fraction. Simplify the answer if possible. $42$ to $28$	2. Write a rate as a fraction. Simplify the answer if possible. $80$ pounds per $6.5$ square inches
3. Find the unit rate. $\text{\$868.80}$ for $24$ hours	4. Marta bought a pack of $6$ paint brushes for $\text{\$32.20}$ . How much did each brush cost? Round to the nearest cent if necessary
5. Find each unit price and then the better buy. Laundry detergent: $64$ ounces for $\text{\$10.99}$ or $48$ ounces for $\text{\$8.49}$	6. Convert a percent to a fraction: $\text{245\%}$
7. Convert a decimal to a percent: $0.07$	8. Convert a fraction to a percent. (Round to $3$ decimal places if needed.) $\frac{11}{8}$
9. What number is $\text{36\%}$ of $450?$	10. $\text{8\%}$ of what number is $\text{\$34}?$
11. $57.6$ is what percent of $360?$	12. One granola bar has $3$ grams of fiber, which is $\text{12\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?
13. Klaudia is going to Poland and wants to exchange $\text{\$1,400}$ into Polish zlotych. If each dollar is $2.91$ zlotych, how many zlotych will Klaudia get?	14. Solve a proportion: $\frac{24}{x}=\frac{3}{7}$
15. Solve a proportion: $\frac{x}{6}=\frac{9}{24}$	16.Solve a proportion: $\frac{2.4}{1.6}=\frac{t}{6.2}$

Test Answers

1. $\frac{3}{2}$	2. $\frac{160}{13}$	3. $36.20
4. $5.37	5. 64 ounces for $10.99 is the better buy	6. $\frac{49}{20}$
7. 7%	8. 137.5%	9. 162
10. 425	11. 16%	12. 25 grams
13. 4074 zlotych	14. 56	15. 2.25
16. 9.3

CHAPTER 4 Measurement, Perimeter, Area, and Volume

Note the many individual shapes in this building.

An intricate class ceiling. The structure is made up of individual shapes.

We are surrounded by all sorts of geometry. Architects use geometry to design buildings. Artists create vivid images out of colorful geometric shapes. Street signs, automobiles, and product packaging all take advantage of geometric properties. In this chapter, we will begin with learning about two measurement systems used in Canada and then we will explore geometry and solve problems related to everyday situations.

Attributions

Calatrava fantasy by Bert Kaufmann is under a CC BY 4.0 Licence.

This chapter has been adapted from the “Introduction” in Chapter 9 of Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

4.1 Systems of Measurement

Learning Objectives

By the end of this section, you will be able to:

Make unit conversions in the imperial system
Use mixed units of measurement in the imperial system
Make unit conversions in the metric system
Use mixed units of measurement in the metric system
Convert between the imperial and the metric systems of measurement
Convert between Fahrenheit and Celsius temperatures

Make Unit Conversions in the Imperial System

There are two systems of measurement commonly used around the world. Most countries use the metric system. Canada uses the metric system, and the United States use the imperial system of measurement. However, people in Canada often use imperial measurements as well. We will look at the imperial system first.

The imperial system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the imperial system and the metric system measure time in seconds, minutes, and hours.

The equivalencies of measurements are shown in the table below. The table also shows, in parentheses, the common abbreviations for each measurement.

Imperial System of Measurement
$\begin{array}{ccc}\mathbf{\text{Length}}\hfill & & \begin{array}{ccc}1\phantom{\rule{0.2em}{0ex}}\text{foot}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill & =\hfill & 12\phantom{\rule{0.2em}{0ex}}\text{inches}\phantom{\rule{0.2em}{0ex}}\text{(in.)}\hfill \\ 1\phantom{\rule{0.2em}{0ex}}\text{yard}\phantom{\rule{0.2em}{0ex}}\text{(yd.)}\hfill & =\hfill & 3\phantom{\rule{0.2em}{0ex}}\text{feet}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill \\ 1\phantom{\rule{0.2em}{0ex}}\text{mile}\phantom{\rule{0.2em}{0ex}}\text{(mi.)}\hfill & =\hfill & 5,280\phantom{\rule{0.2em}{0ex}}\text{feet}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill \end{array}\hfill \end{array}$	$\begin{array}{ccc}\mathbf{\text{Volume}}\hfill & & \begin{array}{ccc}3\phantom{\rule{0.2em}{0ex}}\text{teaspoons}\phantom{\rule{0.2em}{0ex}}\text{(t)}\hfill & =\hfill & 1\phantom{\rule{0.2em}{0ex}}\text{tablespoon}\phantom{\rule{0.2em}{0ex}}\text{(T)}\hfill \\ \text{16 tablespoons}\phantom{\rule{0.2em}{0ex}}\text{(T)}\hfill & =\hfill & \text{1 cup}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill \\ \text{1 cup}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill & =\hfill & \text{8 fluid ounces}\phantom{\rule{0.2em}{0ex}}\text{(fl. oz.)}\hfill \\ \text{1 pint}\phantom{\rule{0.2em}{0ex}}\text{(pt.)}\hfill & =\hfill & \text{2 cups}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill \\ \text{1 quart}\phantom{\rule{0.2em}{0ex}}\text{(qt.)}\hfill & =\hfill & \text{2 pints}\phantom{\rule{0.2em}{0ex}}\text{(pt.)}\hfill \\ \text{1 gallon}\phantom{\rule{0.2em}{0ex}}\text{(gal)}\hfill & =\hfill & \text{4 quarts}\phantom{\rule{0.2em}{0ex}}\text{(qt.)}\hfill \end{array}\hfill \end{array}$
$\begin{array}{ccc}\mathbf{\text{Weight}}\hfill & & \begin{array}{ccc}\text{1 pound}\phantom{\rule{0.2em}{0ex}}\text{(lb.)}\hfill & =\hfill & \text{16 ounces}\phantom{\rule{0.2em}{0ex}}\text{(oz.)}\hfill \\ \text{1 ton}\hfill & =\hfill & \text{2000 pounds}\phantom{\rule{0.2em}{0ex}}\text{(lb.)}\hfill \end{array}\hfill \end{array}$	$\begin{array}{ccc}\mathbf{\text{Time}}\hfill & & \phantom{\rule{1em}{0ex}}\begin{array}{ccc}\text{1 minute}\phantom{\rule{0.2em}{0ex}}\text{(min)}\hfill & =\hfill & \text{60 seconds}\phantom{\rule{0.2em}{0ex}}\text{(sec)}\hfill \\ \text{1 hour}\phantom{\rule{0.2em}{0ex}}\text{(hr)}\hfill & =\hfill & \text{60 minutes}\phantom{\rule{0.2em}{0ex}}\text{(min)}\hfill \\ \text{1 day}\hfill & =\hfill & \text{24 hours}\phantom{\rule{0.2em}{0ex}}\text{(hr)}\hfill \\ \text{1 week}\phantom{\rule{0.2em}{0ex}}\text{(wk)}\hfill & =\hfill & \text{7 days}\hfill \\ \text{1 year}\phantom{\rule{0.2em}{0ex}}\text{(yr)}\hfill & =\hfill & \text{365 days}\hfill \end{array}\hfill \end{array}$

In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.

Identity Property of Multiplication

For any real number $a$ :

$a\cdot1=a$

$1\cdot a=a$

1 is the multiplicative identity.

To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction $\dfrac{1\text{ foot}}{12\text{ inches}}$ . When we multiply by this fraction we do not change the value, but just change the units.

But $\dfrac{12\text{ inches}}{1\text{ foot}}$ also equals 1. How do we decide whether to multiply by $\dfrac{1\text{ foot}}{12\text{ inches}}$ or $\dfrac{12\text{ inches}}{1\text{ foot}}$ ? We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert $66$ inches to feet, which multiplication will eliminate the inches?

$Two expressions are given: 66 inches times the fraction (1 foot) over (12 inches), and 66 inches times the fraction (12 inches) over (1 foot). This second expression is crossed out. Below this, it is stated that “The first form works since 66 inches times the fraction (1 foot) over (12 inches), with inches crossed off in both instances.$

The inches divide out and leave only feet. The second form does not have any units that will divide out and so will not help us.

EXAMPLE 1

MaryAnne is 66 inches tall. Convert her height into feet.

Solution

$A table is given with three columns. In the first column are directions. The second column has exposition, and the third column has the mathematical steps. In the first row, the direction is “Step 1. Multiply the measurement to be converted by; write as a fraction relating the units given and the units needed.” The exposition is “Multiply inches by, writing as a fraction relating inches and feet. We need inches in the denominator so that the inches will divide out!” The mathematical step is 66 inches times the fraction (1 foot) over (12 inches).$ $In the following row, we have “Step 2. Multiply.” The hint is “Think of 66 inches as the quantity 66 inches divided by 1.” The math portion is the fraction (66 inches times 1 foot) over 12 inches.$ $In the following row, we have “Step 3. Simplify the fraction.” The hint is that “Notice: inches divide out.” We obtain 66 feet divided by 12.$ Then the last step is “Step 4. Simplify.” The hint is “Divide 66 by 12.” Hence, our final mathematical statement is 5.5 feet.

TRY IT 1.1

Lexie is 30 inches tall. Convert her height to feet.

Answer

2.5 feet

TRY IT 1.2

Rene bought a hose that is 18 yards long. Convert the length to feet.

Answer

54 feet

HOW TO: Make unit conversions

Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
Multiply.
Simplify the fraction.
Simplify.

When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.

EXAMPLE 2

A female orca in the Salish Sea weighs almost 3.2 tons. Convert her weight to pounds.

Solution

We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction $\dfrac{2000\text{ pounds}}{1\text{ ton}}$ .

	$\text{3.2 tons}$
Multiply the measurement to be converted, by 1.	$\text{3.2 tons}\cdot 1$
Write 1 as a fraction relating tons and pounds.	$\text{3.2 tons}\cdot \dfrac{\text{2,000 pounds}}{\text{1 ton}}$
Simplify.
Multiply.	$\text{6,400 pounds}$
	The female orca weighs almost 6,400 pounds.

TRY IT 2.1

Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.

Answer

8,600 pounds

TRY IT 2.2

The Carnival Destiny cruise ship weighs 51,000 tons. Convert the weight to pounds.

Answer

102,000,000 pounds

Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

EXAMPLE 3

Juliet is going with her family to their summer home. She will be away from her boyfriend for 9 weeks. Convert the time to minutes.

Solution
To convert weeks into minutes we will convert weeks into days, days into hours, and then hours into minutes. To do this we will multiply by conversion factors of 1.

	9 weeks
Write 1 as $\frac{\text{7 days}}{\text{1 week}}$ , $\frac{\text{24 hours}}{\text{1 day}}$ , and $\frac{\text{60 minutes}}{\text{1 hour}}$ .	$\dfrac{\text{9 wk}}{1}\cdot \dfrac{\text{7 days}}{\text{1 wk}}\cdot \dfrac{\text{24 hr}}{\text{1 day}}\cdot \dfrac{\text{60 min}}{\text{1 hr}}$
Divide out the common units.	$\dfrac{\text{9 \cancel{ wk}}}{1}\cdot \dfrac{\text{7\cancel{ days}}}{\text{1\cancel{ wk}}}\cdot \dfrac{\text{24 \cancel{hr}}}{\text{1\cancel{ day}}}\cdot \dfrac{\text{60 min}}{\text{1 \cancel{hr}}}$
Multiply.	$\dfrac{9\cdot 7\cdot 24\cdot \text{60 min}}{1\cdot 1\cdot 1\cdot 1}$
Multiply.	$\text{90,720 min}$

Juliet and her boyfriend will be apart for 90,720 minutes (although it may seem like an eternity!).

TRY IT 3.1

The distance between the earth and the moon is about 250,000 miles. Convert this length to yards.

Answer

440,000,000 yards

TRY IT 3.2

The astronauts of Expedition 28 on the International Space Station spend 15 weeks in space. Convert the time to minutes.

Answer

151,200 minutes

EXAMPLE 4

How many ounces are in 1 gallon?

Solution

We will convert gallons to ounces by multiplying by several conversion factors. Refer to the table on Imperial Systems of Measurement.

	1 gallon
Multiply the measurement to be converted by 1.	$\dfrac{\text{1 gallon}}{1}\cdot \dfrac{\text{4 quarts}}{\text{1 gallon}}\cdot \dfrac{\text{2 pints}}{\text{1 quart}}\cdot \dfrac{\text{2 cups}}{\text{1 pint}}\cdot \dfrac{\text{8 ounces}}{\text{1 cup}}$
Use conversion factors to get to the right unit. Simplify.	$\dfrac{\text{1\cancel{ gallon}}}{1}\cdot \dfrac{\text{4\cancel{ quarts}}}{\text{1\cancel{ gallon}}}\cdot \dfrac{\text{2 \cancel{pints}}}{\text{1 \cancel{quart}}}\cdot \dfrac{\text{2 \cancel{cups}}}{\text{1 \cancel{pint}}}\cdot \dfrac{\text{8 ounces}}{\text{1 \cancel{cup}}}$
Multiply.	$\dfrac{1\cdot 4\cdot 2\cdot 2\cdot \text{8 ounces}}{1\cdot 1\cdot 1\cdot 1\cdot 1}$
Simplify.	$\text{128 ounces}$ There are 128 ounces in a gallon.

TRY IT 4.1

How many cups are in 1 gallon?

Answer

16 cups

TRY IT 4.2

How many teaspoons are in 1 cup?

Answer

48 teaspoons

Use Mixed Units of Measurement in the Imperial System

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!

EXAMPLE 5

Seymour bought three steaks for a barbecue. Their weights were 14 ounces; 1 pound, 2 ounces; and 1 pound, 6 ounces. How many total pounds of steak did he buy?

Solution

We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds.
Convert 22 ounces to pounds and ounces.	1 pound, 6 ounces
Add the pounds and ounces.	2 pounds + 1 pound + 6 ounces
Answer	Seymour bought 3 pounds 6 ounces of steak.

TRY IT 5.1

Laura gave birth to triplets weighing 3 pounds 3 ounces, 3 pounds 3 ounces, and 2 pounds 9 ounces. What was the total birth weight of the three babies?

Answer

9 lbs. 8 oz

TRY IT 5.2

Stan cut two pieces of crown molding for his family room that were 8 feet 7 inches and 12 feet 11 inches. What was the total length of the molding?

Answer

21 ft. 6 in.

EXAMPLE 6

Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?

Solution

We will multiply the length of one plank to find the total length.

Multiply the inches and then the feet.
Convert the 16 inches to feet. Add the feet.
	Anthony bought 25 feet and 4 inches of wood.

TRY IT 6.1

Henri wants to triple his vegan spaghetti sauce recipe that uses 1 pound 8 ounces of black beans. How many pounds of black beans will he need?

Answer

4 lbs. 8 oz.

TRY IT 6.2

Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?

Answer

11 gallons 2 qt.

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a metre. One kilometre is 1,000 metres; the prefix kilo means thousand. One centimetre is $\dfrac{1}{100}$ of a metre, just like one cent is $\dfrac{1}{100}$ of one dollar.

The equivalencies of measurements in the metric system are shown in the table below. The common abbreviations for each measurement are given in parentheses.

Metric System of Measurement
Length	Mass	Capacity
1 kilometre (km) = 1,000 m 1 hectometre (hm) = 100 m 1 dekametre (dam) = 10 m 1 metre (m) = 1 m 1 decimetre (dm) = 0.1 m 1 centimetre (cm) = 0.01 m 1 millimetre (mm) = 0.001 m	1 kilogram (kg) = 1,000 g 1 hectogram (hg) = 100 g 1 dekagram (dag) = 10 g 1 gram (g) = 1 g 1 decigram (dg) = 0.1 g 1 centigram (cg) = 0.01 g 1 milligram (mg) = 0.001 g	1 kilolitre (kL) = 1,000 L 1 hectolitre (hL) = 100 L 1 dekalitre (daL) = 10 L 1 litre (L) = 1 L 1 decilitre (dL) = 0.1 L 1 centilitre (cL) = 0.01 L 1 millilitre (mL) = 0.001 L
1 metre = 100 centimetres 1 metre = 1,000 millimetres	1 gram = 100 centigrams 1 gram = 1,000 milligrams	1 litre = 100 centilitre s 1 litre = 1,000 millilitre s

To make conversions in the metric system, we will use the same technique we did in the Imperial system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.

Have you ever run a 5K or 10K race? The length of those races are measured in kilometres. The metric system is commonly used in Canada when talking about the length of a race.

EXAMPLE 7

Nick ran a 10K race. How many metres did he run?

Solution

We will convert kilometres to metres using the identity property of multiplication.

	10 kilometres
Multiply the measurement to be converted by 1.	10 kilometres × 1
Write 1 as a fraction relating kilometres and metres.	$10\text{ kilometres}\times\dfrac{1,000\text{ metres}}{1\text{ kilometres}}$
Simplify.	$10\text{ \cancel{kilometres}}\times\dfrac{1,000\text{ metres}}{1\text{ \cancel{kilometres}}}$
Multiply.	10,000 metres
	Nick ran 10,000 metres.

TRY IT 7.1

Sandy completed her first 5K race! How many metres did she run?

Answer

5,000 metres

TRY IT 7.2

Herman bought a rug 2.5 metres in length. How many centimetres is the length?

Answer

250 centimetres

EXAMPLE 8

Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?

Solution

We will convert grams into kilograms.


Multiply the measurement to be converted by 1.
Write 1 as a function relating kilograms and grams.
Simplify.
Multiply.	$\dfrac{\text{3,200 kilograms}}{1,000}$
Divide.	3.2 kilograms The baby weighed 3.2 kilograms.

TRY IT 8.1

Kari’s newborn baby weighed 2,800 grams. How many kilograms did the baby weigh?

Answer

2.8 kilograms

TRY IT 8.2

Anderson received a package that was marked 4,500 grams. How many kilograms did this package weigh?

Answer

4.5 kilograms

As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.

To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.

We can apply this pattern when we make measurement conversions in the metric system. In Example 8, we changed 3,200 grams to kilograms by multiplying by $\dfrac{1}{1000}$ (or 0.001). This is the same as moving the decimal three places to the left.

$We have the statement 3200 g times the fraction 1 kg over 1000 g, with the g’s crossed out. Below this, we have 3.2. We also have the statement 3200 times 1/1000, with an arrow drawn from the right of the final 0 in 3200 to the space between the 0’s, to the space between the 2 and the 0, and then to the space between the 3 and the 2. Below this, we have 3.2.$

Figure.1

EXAMPLE 9

Convert a) 350 L to kilolitres b) 4.1 L to millilitre s.

Solution

We will convert litres to kilolitres. In the Metric System of Measurement table, we see that $1\phantom{\rule{0.2em}{0ex}}\text{kilolitre}=\text{1,000 litres.}$

$\text{350 L}$

Multiply by 1, writing 1 as a fraction relating litres to kilolitres. $\text{350 L}\cdot \dfrac{\text{1 kL}}{\text{1,000 L}}$

Simplify. $350\cancel{\text{L}}\cdot \dfrac{\text{1 kL}}{1,000\cancel{\text{L}}}$

Move the decimal 3 units to the left. $\text{0.35 kL}$
We will convert litres to millilitre s. From Metric System of Measurement table we see that $\text{1 litre}=\text{1,000 millilitre s.}$

Multiply by 1, writing 1 as a fraction relating litres to millilitre s.

Simplify.

Move the decimal 3 units to the right.

TRY IT 9.1

Convert: a) 725 L to kilolitres b) 6.3 L to millilitres

Answer

a) 0.725 kilolitres b) 6,300 millilitres

TRY IT 9.2

Convert: a) 350 hL to litres b) 4.1 L to centilitres

Answer

a) 35,000 litres b) 410 centilitres

Use Mixed Units of Measurement in the Imperial System

Performing arithmetic operations on measurements with mixed units of measures in the imperial system requires the same care we used in the Canadian system. Make sure to add or subtract like units.

EXAMPLE 10

Ryland is 1.6 metres tall. His younger brother is 85 centimetres tall. How much taller is Ryland than his younger brother?

Solution

We can convert both measurements to either centimetres or metres. Since metres is the larger unit, we will subtract the lengths in metres. We convert 85 centimetres to metres by moving the decimal 2 places to the left.

Write the 85 centimetres as metres.

$\begin{array}{r}1.60\text{m}\\ \underline{-0.85\text{m}} \\ 0.75\text{m}\hfill \end{array}$

Ryland is 0.75 m taller than his brother.

TRY IT 10.1

Mariella is 1.58 metres tall. Her daughter is 75 centimetres tall. How much taller is Mariella than her daughter? Write the answer in centimetres.

Answer

83 centimetres

TRY IT 10.2

The fence around Hank’s yard is 2 metres high. Hank is 96 centimetres tall. How much shorter than the fence is Hank? Write the answer in metres.

Answer

1.04 metres

EXAMPLE 11

Dena’s recipe for lentil soup calls for 150 millilitre s of olive oil. Dena wants to triple the recipe. How many litres of olive oil will she need?

Solution

We will find the amount of olive oil in millileters then convert to litres.

	Triple 150 mL
Translate to algebra.	$3 \cdot 150 \text{ mL}$
Multiply.	450 mL
Convert to litres.	$450\cdot \dfrac{0.001\text{ L}}{1\text{ mL}}$
Simplify.	0.45 L
	Dena needs 0.45 litres of olive oil.

TRY IT 11.1

A recipe for Alfredo sauce calls for 250 millilitre s of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by 8. How many litres of milk will she need?

Answer

2 litres

TRY IT 11.2

To make one pan of baklava, Dorothea needs 400 grams of filo pastry. If Dorothea plans to make 6 pans of baklava, how many kilograms of filo pastry will she need?

Answer

2.4 kilograms

Convert Between the Imperial and the Metric Systems of Measurement

Many measurements in Canada are made in metric units. Our soda may come in 2-litre bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.

The table below shows some of the most common conversions.

Conversion Factors Between Imperial and Metric Systems
Length	Mass	Capacity
$\begin{array}{c}\begin{array}{ccc}\text{1 in.}\hfill & =\hfill & \text{2.54 cm}\hfill \\ \text{1 ft.}\hfill & =\hfill & \text{0.305 m}\hfill \\ \text{1 yd.}\hfill & =\hfill & \text{0.914 m}\hfill \\ \text{1 mi.}\hfill & =\hfill & \text{1.609 km}\hfill \\ \text{1 m}\hfill & =\hfill & \text{3.28 ft.}\hfill \end{array}\hfill \end{array}$	$\begin{array}{c}\begin{array}{ccc}\text{1 lb.}\hfill & =\hfill & \text{0.45 kg}\hfill \\ \text{1 oz.}\hfill & =\hfill & \text{28 g}\hfill \\ \text{1 kg}\hfill & =\hfill & \text{2.2 lb.}\hfill \end{array}\hfill \end{array}$	$\begin{array}{c}\begin{array}{ccc}\text{1 qt.}\hfill & =\hfill & \text{0.95 L}\hfill \\ \text{1 fl. oz.}\hfill & =\hfill & \text{30 mL}\hfill \\ \text{1 L}\hfill & =\hfill & \text{1.06 qt.}\hfill \end{array}\hfill \end{array}$

(Figure.2) shows how inches and centimetres are related on a ruler.

Figure.2

(Figure.3) shows the ounce and millilitre markings on a measuring cup.

A measuring cup showing millilitre s and ounces.

Figure.3

(Figure.4) shows how pounds and kilograms marked on a bathroom scale.

We are given an image of a bathroom scale showing pounds.

Figure.4

We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.

EXAMPLE 12

Lee’s water bottle holds 500 mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.

Solution

	500 mL
Multiply by a unit conversion factor relating mL and ounces.	$500\text{ millilitres}\cdot \dfrac{1\text{ ounce}}{30\text{ millilitres}}$
Simplify.	$\dfrac{\text{500 ounce}}{30}$
Divide.	16.7 ounces.
	The water bottle has 16.7 ounces.

TRY IT 12.1

How many quarts of soda are in a 2-L bottle? Round your answer to nearest tenth.

Answer

2.1 quarts

TRY IT 12.2

How many litres are in 4 quarts of milk?

Answer

3.8 litres

EXAMPLE 13

Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometres. How many miles until the next rest stop? Round your answer to nearest mile.

Solution

	100 kilometres
Multiply by a unit conversion factor relating km and mi.	$100\text{ kilometres}\cdot \dfrac{1\text{ mile}}{1.609\text{ kilometre}}$
Simplify.	$\dfrac{\text{100 miles}}{1.609}$
Divide.	62 ounces.
	Soleil will travel 62 miles.

TRY IT 13.1

The height of Mount Kilimanjaro is 5,895 metres. Convert the height to feet.

Answer

19,335.6 feet

TRY IT 13.2

The flight distance from Toronto to Vancouver is 3,364 kilometres. Convert the distance to miles. Round your answer to nearest mile.

Answer

2,090 miles

Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for $71$ °F what does that mean?

The Canadian and imperial systems use different scales to measure temperature. The Canadian system uses degrees Celsius, written °C. The imperial system uses degrees Fahrenheit, written°F. (Figure.5) shows the relationship between the two systems.

The diagram shows normal body temperature, along with the freezing and boiling temperatures of water in degrees Fahrenheit and degrees Celsius.

Two thermometres are shown, one in Celsius (°C) and another in Fahrenheit (°F). They are marked “Water boils” at 100°C and 212°F. They are marked “Normal body temperature” at 37°C and 98.6°F. They are marked “Water freezes” at 0°C and 32°F.

Figure.5

Temperature Conversion

To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula

$C=\dfrac{5}{9}\left(F-32\right)$ .

To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula

$F=\dfrac{9}{5}C+32$ .

EXAMPLE 14

Convert 50° Fahrenheit into degrees Celsius.

Solution

We will substitute 50°F into the formula to find C.



Simplify in parentheses.
Multiply.
	So we found that 50°F is equivalent to 10°C.

TRY IT 14.1

Convert the Fahrenheit temperature to degrees Celsius: 59° Fahrenheit.

Answer

15°C

TRY IT 14.2

Convert the Fahrenheit temperature to degrees Celsius: 41° Fahrenheit.

Answer

5°C

EXAMPLE 15

While visiting Paris, Woody saw the temperature was 20° Celsius. Convert the temperature into degrees Fahrenheit.

Solution

We will substitute 20°C into the formula to find F.



Multiply.
Add.
	So we found that 20°C is equivalent to 68°F.

TRY IT 15.1

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Helsinki, Finland, was 15° Celsius.

Answer

59°F

TRY IT 15.2

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Sydney, Australia, was 10° Celsius.

Answer

50° F

Key Concepts

Metric System of Measurement
- Length
  $\begin{array}{lcl}\text{1 kilometre (km)} & = & \text{1,000 m} \\ \text{1 hectometre (hm)} & = & \text{100 m} \\ \text{1 dekametre (dam)} & = & \text{10 m} \\ \text{1 metre (m)} & = & \text{1 m} \\ \text{1 decimetre (dm)} & = & \text{0.1 m} \\ \text{1 centimetre (cm)} & = & \text{0.01 m}\hfill \\ \text{1 millimetre (mm)} & = & \text{0.001 m} \\ \text{1 metre} & = & \text{100 centimetres} \\ \text{1 metre} & = & \text{1,000 millimetres} \end{array}$
- Mass
  $\begin{array}{lcl}\text{1 kilogram (kg)} & = & \text{1,000 g} \\ \text{1 hectogram (hg)} & = & \text{100 g} \\ \text{1 dekagram (dag)} & = & \text{10 g} \\ \text{1 gram (g)} & = & \text{1 g} \\ \text{1 decigram (dg)} & = & \text{0.1 g} \\ \text{1 centigram (cg)} & = & \text{0.01 g} \\ \text{1 milligram (mg)} & = & \text{0.001 g} \\ \text{1 gram} & = & \text{100 centigrams} \\ \text{1 gram} & = & \text{1,000 milligrams} \end{array}$
- Capacity
  $\begin{array}{lcl}\text{1 kilolitre (kL)} & = & \text{1,000 L} \\ \text{1 hectolitre (hL)} & = & \text{100 L} \\ \text{1 dekalitre (daL)} & = & \text{10 L} \\ \text{1 litre (L)} & = & \text{1 L} \\ \text{1 decilitre (dL)} & = & \text{0.1 L} \\ \text{1 centilitre (cL)} & = & \text{0.01 L} \\ \text{1 millilitre (mL)} & = & \text{0.001 L} \\ \text{1 litre} & = & \text{100 centilitre s} \\ \text{1 litre} & = & \text{1,000 millilitre s}\end{array}$
Temperature Conversion
- To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula $\text{C}=\dfrac{5}{9}\left(\text{F}-32\right)$
- To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula $\text{F}=\dfrac{9}{5}\text{C}+32$

Practice Makes Perfect

Make Unit Conversions in the Imperial System

In the following exercises, convert the units.

1. A park bench is 6 feet long. Convert the length to inches.	2. A floor tile is 2 feet wide. Convert the width to inches.
3. A ribbon is 18 inches long. Convert the length to feet.	4. Carson is 45 inches tall. Convert his height to feet.
5. A football field is 160 feet wide. Convert the width to yards.	6. On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet.
7. Ulises lives 1.5 miles from school. Convert the distance to feet.	8. Denver, Colorado, is 5,183 feet above sea level. Convert the height to miles.
9. A killer whale weighs 4.6 tons. Convert the weight to pounds.	10. Blue whales can weigh as much as 150 tons. Convert the weight to pounds.
11. An empty bus weighs 35,000 pounds. Convert the weight to tons.	12. At take-off, an airplane weighs 220,000 pounds. Convert the weight to tons.
13. Rocco waited $1\dfrac{1}{2}$ hours for his appointment. Convert the time to seconds.	14. Misty’s surgery lasted $2\dfrac{1}{4}$ hours. Convert the time to seconds.
15. How many teaspoons are in a pint?	16. How many tablespoons are in a gallon?
17. JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces.	18. April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces.
19. Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons.	20. Lance needs 50 cups of water for the runners in a race. Convert the volume to gallons.
21. Jon is 6 feet 4 inches tall. Convert his height to inches.	22. Faye is 4 feet 10 inches tall. Convert her height to inches.
23. The voyage of the Mayflower took 2 months and 5 days. Convert the time to days.	24. Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours.
25. Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces.	26. Baby Audrey weighted 6 pounds 15 ounces at birth. Convert her weight to ounces.

Use Mixed Units of Measurement in the Imperial System

In the following exercises, solve.

27. Eli caught three fish. The weights of the fish were 2 pounds 4 ounces, 1 pound 11 ounces, and 4 pounds 14 ounces. What was the total weight of the three fish?	28. Judy bought 1 pound 6 ounces of almonds, 2 pounds 3 ounces of walnuts, and 8 ounces of cashews. How many pounds of nuts did Judy buy?
29. One day Anya kept track of the number of minutes she spent driving. She recorded 45, 10, 8, 65, 20, and 35. How many hours did Anya spend driving?	30. Last year Eric went on 6 business trips. The number of days of each was 5, 2, 8, 12, 6, and 3. How many weeks did Eric spend on business trips last year?
31. Renee attached a 6 feet 6 inch extension cord to her computer’s 3 feet 8 inch power cord. What was the total length of the cords?	32. Fawzi’s SUV is 6 feet 4 inches tall. If he puts a 2 feet 10 inch box on top of his SUV, what is the total height of the SUV and the box?
33. Leilani wants to make 8 placemats. For each placemat she needs 18 inches of fabric. How many yards of fabric will she need for the 8 placemats?	34. Mireille needs to cut 24 inches of ribbon for each of the 12 girls in her dance class. How many yards of ribbon will she need altogether?

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

35. Ghalib ran 5 kilometres. Convert the length to metres.	36. Kitaka hiked 8 kilometres. Convert the length to metres.
37. Estrella is 1.55 metres tall. Convert her height to centimetres.	38. The width of the wading pool is 2.45 metres. Convert the width to centimetres.
39. Mount Whitney is 3,072 metres tall. Convert the height to kilometres.	40. The depth of the Mariana Trench is 10,911 metres. Convert the depth to kilometres.
41. June’s multivitamin contains 1,500 milligrams of calcium. Convert this to grams.	42. A typical ruby-throated hummingbird weights 3 grams. Convert this to milligrams.
43. One stick of butter contains 91.6 grams of fat. Convert this to milligrams.	44. One serving of gourmet ice cream has 25 grams of fat. Convert this to milligrams.
45. The maximum mass of an airmail letter is 2 kilograms. Convert this to grams.	46. Dimitri’s daughter weighed 3.8 kilograms at birth. Convert this to grams.
47. A bottle of wine contained 750 millilitre s. Convert this to litres.	48. A bottle of medicine contained 300 millilitre s. Convert this to litres.

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve.

49. Matthias is 1.8 metres tall. His son is 89 centimetres tall. How much taller is Matthias than his son?	50. Stavros is 1.6 metres tall. His sister is 95 centimetres tall. How much taller is Stavros than his sister?
51. A typical dove weighs 345 grams. A typical duck weighs 1.2 kilograms. What is the difference, in grams, of the weights of a duck and a dove?	52. Concetta had a 2-kilogram bag of flour. She used 180 grams of flour to make biscotti. How many kilograms of flour are left in the bag?
53. Harry mailed 5 packages that weighed 420 grams each. What was the total weight of the packages in kilograms?	54. One glass of orange juice provides 560 milligrams of potassium. Linda drinks one glass of orange juice every morning. How many grams of potassium does Linda get from her orange juice in 30 days?
55. Jonas drinks 200 millilitre s of water 8 times a day. How many litres of water does Jonas drink in a day?	56. One serving of whole grain sandwich bread provides 6 grams of protein. How many milligrams of protein are provided by 7 servings of whole grain sandwich bread?

Convert Between the Imperial and the Metric Systems of Measurement

In the following exercises, make the unit conversions. Round to the nearest tenth.

57. Bill is 75 inches tall. Convert his height to centimetres.	58. Frankie is 42 inches tall. Convert his height to centimetres.
59. Marcus passed a football 24 yards. Convert the pass length to metres	60. Connie bought 9 yards of fabric to make drapes. Convert the fabric length to metres.
61. According to research conducted by the CRC, Canadians regrettably produce more garbage per capita than any other country on earth, at 2,172.6 pounds per person annually. Convert the waste to kilograms.	62. An average Canadian will throw away 163,000 pounds of trash over his or her lifetime. Convert this weight to kilograms.
63. A 5K run is 5 kilometres long. Convert this length to miles.	64. Kathryn is 1.6 metres tall. Convert her height to feet.
65. Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds.	66. Jackson’s backpack weighed 15 kilograms. Convert the weight to pounds.
67. Ozzie put 14 gallons of gas in his truck. Convert the volume to litres.	68. Bernard bought 8 gallons of paint. Convert the volume to litres.

Convert between Fahrenheit and Celsius Temperatures

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

69. 86° Fahrenheit	70. 77° Fahrenheit
71. 104° Fahrenheit	72. 14° Fahrenheit
73. 72° Fahrenheit	74. 4° Fahrenheit
75. 0° Fahrenheit	76. 120° Fahrenheit

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

77. 5° Celsius	78. 25° Celsius
79. -10° Celsius	80. -15° Celsius
81. 22° Celsius	82. 8° Celsius
83. 43° Celsius	84. 16° Celsius

Everyday Math

85. Nutrition Julian drinks one can of soda every day. Each can of soda contains 40 grams of sugar. How many kilograms of sugar does Julian get from soda in 1 year?

86. Reflectors The reflectors in each lane-marking stripe on a highway are spaced 16 yards apart. How many reflectors are needed for a one mile long lane-marking stripe?

Writing Exercises

87. Some people think that 65° to 75° Fahrenheit is the ideal temperature range.

a) What is your ideal temperature range? Why do you think so?

b) Convert your ideal temperatures from Fahrenheit to Celsius.

88.

a) Did you grow up using the Canadian. or the Imperial system of measurement?

b) Describe two examples in your life when you had to convert between the two systems of measurement.

Answers

1. 72 inches	3. 1.5 feet	5. $53\dfrac{1}{3}$ yards
7. 7,920 feet	9. 9,200 pounds	11. $17\dfrac{1}{2}$ tons
13. 5,400 s	15. $96$ teaspoons	17. $224$ ounces
19. $1\dfrac{1}{4}$ gallons	21. 76 in.	23. 65 days
25. 115 ounces	27. 8 lbs. 13 oz.	29. 3.05 hours
31. 10 ft. 2 in.	33. 4 yards	35. 5,000 metres
37. 155 centimetres	39. 3.072 kilometres	41. 1.5 grams
43. 91,600 milligrams	45. 2,000 grams	47. 0.75 litres
49. 91 centimetres	49. 91 centimetres	49. 91 centimetres
53. 2.1 kilograms	55. 1.6 litres	57. 190.5 centimetres
59. 21.9 metres	61. 985.5 kilograms	63. 3.1 miles
65. 44 pounds	67. 53.2 litres	69. 30°C
71. 40°C	73. 22.2°C	75. -17.8°C
77. 41°F	79. 14°F	81. 71.6°F
83. 109.4°F	85. 14.6 kilograms	87. Answers may vary.

Attributions

This chapter has been adapted from “Systems of Measurement” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

4.2 Use Properties of Rectangles, Triangles, and Trapezoids

Learning Objectives

By the end of this section, you will be able to:

Understand linear, square, and cubic measure
Use properties of rectangles
Use properties of triangles
Use properties of trapezoids

Understand Linear, Square, and Cubic Measure

When you measure your height or the length of a garden hose, you use a ruler or tape measure (Figure.1). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimetre and metre are units of linear measure.

This tape measure measures inches along the top and centimetres along the bottom.

A picture of a portion of a tape measure is shown. The top shows the numbers 1 through 5. The portion from the beginning to the 1 has a red circle and an arrow to a picture from 0 to 1 inch, with 1 sixteenth, 1 eighth, 3 eighths, 1 half, and 3 fourths labeled. Above this, it is labeled “Standard Measures.” The bottom of the tape measure shows the numbers 1 through 10, then 1 and 2. The region from the edge to about 3 and a half has a red circle with an arrow pointing to a picture from 0 to 3.5. It is labeled 0, 1 cm, 1.7 cm, 2.3 cm and 3.5 cm. Above this, it is labeled “Metric (S).”

Figure.1

When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the area, a measure of the region needed to cover a surface. Area is measured is square units. We often use square inches, square feet, square centimetres, or square miles to measure area. A square centimetre is a square that is one centimetre (cm) on each side. A square inch is a square that is one inch on each side (Figure.2).

Square measures have sides that are each $1$ unit in length.

Two squares are shown. The smaller one has sides labeled 1 cm and is 1 square centimetre. The larger one has sides labeled 1 inch and is 1 square inch.

Figure.2

(Figure.3) shows a rectangular rug that is $2$ feet long by $3$ feet wide. Each square is $1$ foot wide by $1$ foot long, or $1$ square foot. The rug is made of $6$ squares. The area of the rug is $6$ square feet.

A rectangle is shown. It has 3 squares across and 2 squares down, a total of 6 squares.

Figure 3 The rug contains six squares of 1 square foot each, so the total area of the rug is 6 square feet.

When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring volume. Volume is measured in cubic units such as cubic inches or cubic centimetres. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimetres, cubic inches, and cubic feet. A cubic centimetre is a cube that measures one centimetre on each side, while a cubic inch is a cube that measures one inch on each side (Figure.4).

Two cubes are shown. The smaller one has sides labeled 1 cm and is labeled as 1 cubic centimetre. The larger one has sides labeled 1 inch and is labeled as 1 cubic inch.

Figure 4 Cubic measures have sides that are 1 unit in length.

Suppose the cube in (Figure.5) measures $3$ inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find $27$ little cubes, with each one measuring one inch on all sides. So each little cube has a volume of $1$ cubic inch, and the volume of the big cube is $27$ cubic inches.

A cube that measures 3 inches on each side is made up of 27 one-inch cubes, or 27 cubic inches.

A cube is shown, comprised of smaller cubes. Each side of the cube has 3 smaller cubes across, for a total of 27 smaller cubes.

Figure.5

EXAMPLE 1

For each item, state whether you would use linear, square, or cubic measure:

a) amount of carpeting needed in a room

b) extension cord length

c) amount of sand in a sandbox

d) length of a curtain rod

e) amount of flour in a canister

f) size of the roof of a doghouse.

Solution

a) You are measuring how much surface the carpet covers, which is the area.	square measure
b) You are measuring how long the extension cord is, which is the length.	linear measure
c) You are measuring the volume of the sand.	cubic measure
d) You are measuring the length of the curtain rod.	linear measure
e) You are measuring the volume of the flour.	cubic measure
f) You are measuring the area of the roof.	square measure

TRY IT 1.1

Determine whether you would use linear, square, or cubic measure for each item.

a) amount of paint in a can b) height of a tree c) floor of your bedroom d) diametre of bike wheel e) size of a piece of sod f) amount of water in a swimming pool

Answer

cubic
linear
square
linear
square
cubic

TRY IT 1.2

Determine whether you would use linear, square, or cubic measure for each item.

a) volume of a packing box b) size of patio c) amount of medicine in a syringe d) length of a piece of yarn e) size of housing lot f) height of a flagpole

Answer

cubic
square
cubic
linear
square
linear

Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.

Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.

Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.

Perimeter and Area

The perimeter is a measure of the distance around a figure.

The area is a measure of the surface covered by a figure.

(Figure. 6) shows a square tile that is $1$ inch on each side. If an ant walked around the edge of the tile, it would walk $4$ inches. This distance is the perimeter of the tile.

Since the tile is a square that is $1$ inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape.

$\begin{array}{c}\text{Perimeter}=4\phantom{\rule{0.2em}{0ex}}\text{inches}\\ \text{Area}=1\phantom{\rule{0.2em}{0ex}}\text{square inch}\end{array}$

A 5 square by 5 square checkerboard is shown with each side labeled 1 inch. An image of an ant is shown on the top left square.

Figure 6 When the ant walks completely around the tile on its edge, it is tracing the perimeter of the tile. The area of the tile is 1 square inch.

EXAMPLE 2

Each of two square tiles is $1$ square inch. Two tiles are shown together.

a) What is the perimeter of the figure?

b) What is the area?

A checkerboard is shown. It has 10 squares across the top and 5 down the side.

Solution

a) The perimeter is the distance around the figure. The perimeter is $6$ inches.

b) The area is the surface covered by the figure. There are $2$ square inch tiles so the area is $2$ square inches.

A checkerboard is shown. It has 10 squares across the top and 5 down the side. The top and bottom each have two adjacent 1 inch labels across, the sides have 1 inch labels.

TRY IT 2.1

Find the a) perimeter and b) area of the figure:

A rectangle is shown comprised of 3 squares.

Answer

8 inches
3 sq. inches

TRY IT 2.2

Find the a) perimeter and b) area of the figure:

A square is shown comprised of 4 smaller squares.

Answer

8 centimetres
4 sq. centimetres

Use the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, $L$ , and the adjacent side as the width, $W$ . See (Figure.7).

A rectangle has four sides, and four right angles. The sides are labeled L for length and W for width.

A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W.

Figure.7

The perimeter, $P$ , of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk $L+W+L+W$ units, or two lengths and two widths. The perimeter then is

$\begin{array}{c}P=L+W+L+W\hfill \\ \hfill \text{or}\hfill \\ P=2L+2W\hfill \end{array}$

What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was $2$ feet long by $3$ feet wide, and its area was $6$ square feet. See (Figure.8). Since $A=2\cdot 3$ , we see that the area, $A$ , is the length, $L$ , times the width, $W$ , so the area of a rectangle is $A=L\cdot W$ .

The area of this rectangular rug is $6$ square feet, its length times its width.

A rectangle is shown. It is made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3.

Figure.8

Properties of Rectangles

Rectangles have four sides and four right $\left(\text{90}\right)$ ° angles.
The lengths of opposite sides are equal.
The perimeter, $P$ , of a rectangle is the sum of twice the length and twice the width. See (Figure 7). $P=2L+2W$

The area, $A$ , of a rectangle is the length times the width. $A=L\cdot W$

For easy reference as we work the examples in this section, we will state the Problem Solving Strategy for Geometry Applications here.

HOW TO: Use a Problem Solving Strategy for Geometry Applications

Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for. Choose a variable to represent that quantity.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

EXAMPLE 3

The length of a rectangle is $32$ metres and the width is $20$ metres. Find a) the perimeter, and b) the area.

Solution

a)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it.	Let P = the perimeter
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The perimeter of the rectangle is 104 metres.

b)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a rectangle
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area of the rectangle is 60 square metres.

TRY IT 3.1

The length of a rectangle is $120$ yards and the width is $50$ yards. Find a) the perimeter and b) the area.

Answer

340 yd
6000 sq. yd

TRY IT 3.2

The length of a rectangle is $62$ feet and the width is $48$ feet. Find a) the perimeter and b) the area.

Answer

220 ft
2976 sq. ft

EXAMPLE 4

Find the length of a rectangle with perimeter $50$ inches and width $10$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length of the rectangle
Step 3. Name. Choose a variable to represent it.	Let L = the length
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 15 inches.

TRY IT 4.1

Find the length of a rectangle with a perimeter of $80$ inches and width of $25$ inches.

Answer

15 in.

TRY IT 4.2

Find the length of a rectangle with a perimeter of $30$ yards and width of $6$ yards.

Answer

9 yd

In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

EXAMPLE 5

The width of a rectangle is two inches less than the length. The perimeter is $52$ inches. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width of the rectangle
Step 3. Name. Choose a variable to represent it. Now we can draw a figure using these expressions for the length and width.	Since the width is defined in terms of the length, we let L = length. The width is two feet less that the length, so we let L − 2 = width
Step 4.Translate. Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information. Substitute in the given information.
Step 5. Solve the equation.	$52=2L+2L-4$
Combine like terms.	$52=4L-4$
Add 4 to each side.	$56=4L$
Divide by 4.	$\frac{56}{4}=\frac{4L}{4}$
	$14=L$
	The length is 14 inches.
Now we need to find the width.
The width is L − 2.	The width is 12 inches.
Step 6. Check: Since $14+12+14+12=52$ , this works!
Step 7. Answer the question.	The length is 14 feet and the width is 12 feet.

TRY IT 5.1

The width of a rectangle is seven metres less than the length. The perimeter is $58$ metres. Find the length and width.

Answer

18 m, 11 m

TRY IT 5.2

The length of a rectangle is eight feet more than the width. The perimeter is $60$ feet. Find the length and width.

Answer

11 ft , 19 ft

EXAMPLE 6

The length of a rectangle is four centimetres more than twice the width. The perimeter is $32$ centimetres. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width
Step 3. Name. Choose a variable to represent it.	let W = width The length is four more than twice the width. 2w + 4 = length
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 12 cm and the width is 4 cm.

TRY IT 6.1

The length of a rectangle is eight more than twice the width. The perimeter is $64$ feet. Find the length and width.

Answer

8 ft, 24 ft

TRY IT 6.2

The width of a rectangle is six less than twice the length. The perimeter is $18$ centimetres. Find the length and width.

Answer

5 cm, 4 cm

EXAMPLE 7

The area of a rectangular room is $168$ square feet. The length is $14$ feet. What is the width?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the width of a rectangular room
Step 3. Name. Choose a variable to represent it.	Let W = width
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The width of the room is 12 feet.

TRY IT 7.1

The area of a rectangle is $598$ square feet. The length is $23$ feet. What is the width?

Answer

26 ft

TRY IT 7.2

The width of a rectangle is $21$ metres. The area is $609$ square metres. What is the length?

Answer

29 m

EXAMPLE 8

The perimeter of a rectangular swimming pool is $150$ feet. The length is $15$ feet more than the width. Find the length and width.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length and width of the pool
Step 3. Name. Choose a variable to represent it. The length is 15 feet more than the width.	Let $W=\text{width}$ $W+15=\text{length}$
Step 4.Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the pool is 45 feet and the width is 30 feet.

TRY IT 8.1

The perimeter of a rectangular swimming pool is $200$ feet. The length is $40$ feet more than the width. Find the length and width.

Answer

30 ft, 70 ft

TRY IT 8.2

The length of a rectangular garden is $30$ yards more than the width. The perimeter is $300$ yards. Find the length and width.

Answer

60 yd, 90 yd

Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in (Figure.9), we’ve labeled the length $b$ and the width $h$ , so it’s area is $bh$ .

The area of a rectangle is the base, $b$ , times the height, $h$ .

A rectangle is shown. The side is labeled h and the bottom is labeled b. The centre says A equals bh.

Figure.9

We can divide this rectangle into two congruent triangles (Figure.10). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or $\frac{1}{2}bh$ . This example helps us see why the formula for the area of a triangle is $A=\frac{1}{2}bh$ .

A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.

A rectangle is shown. A diagonal line is drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says “Area of each triangle,” and shows the equation A equals one-half bh.

Figure.10

The formula for the area of a triangle is $A=\frac{1}{2}bh$ , where $b$ is the base and $h$ is the height.

To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a $\text{90}$ ° angle with the base. (Figure.11) shows three triangles with the base and height of each marked.

The height $h$ of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a $\text{90}$ ° angle with the base.

Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.

Figure.11

Triangle Properties

For any triangle $\Delta ABC$ , the sum of the measures of the angles is $\text{180}$ °.

$m\angle A+m\angle B+m\angle C=\text{180}$ °

The perimeter of a triangle is the sum of the lengths of the sides.

$P=a+b+c$

The area of a triangle is one-half the base, $b$ , times the height, $h$ .

$A=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}bh$

A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.

EXAMPLE 9

Find the area of a triangle whose base is $11$ inches and whose height is $8$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the triangle
Step 3. Name. Choose a variable to represent it.	let A = area of the triangle
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area is 44 square inches.

TRY IT 9.1

Find the area of a triangle with base $13$ inches and height $2$ inches.

Answer

13 sq. in.

TRY IT 9.2

Find the area of a triangle with base $14$ inches and height $7$ inches.

Answer

49 sq. in.

EXAMPLE 10

The perimeter of a triangular garden is $24$ feet. The lengths of two sides are $4$ feet and $9$ feet. How long is the third side?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.	Let c = the third side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side is 11 feet long.

TRY IT 10.1

The perimeter of a triangular garden is $48$ feet. The lengths of two sides are $18$ feet and $22$ feet. How long is the third side?

Answer

8 ft

TRY IT 10.2

The lengths of two sides of a triangular window are $7$ feet and $5$ feet. The perimeter is $18$ feet. How long is the third side?

Answer

6 ft

EXAMPLE 11

The area of a triangular church window is $90$ square metres. The base of the window is $15$ metres. What is the window’s height?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	height of a triangle
Step 3. Name. Choose a variable to represent it.	Let h = the height
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The height of the triangle is 12 metres.

TRY IT 11.1

The area of a triangular painting is $126$ square inches. The base is $18$ inches. What is the height?

Answer

14 in.

TRY IT 11.2

A triangular tent door has an area of $15$ square feet. The height is $5$ feet. What is the base?

Answer

6 ft

Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. (Figure.12) shows both types of triangles.

In an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.

Two triangles are shown. All three sides of the triangle on the left are labeled s. It is labeled “equilateral triangle”. Two sides of the triangle on the right are labeled s. It is labeled “isosceles triangle”.

Figure.12

Isosceles and Equilateral Triangles

An isosceles triangle has two sides the same length.

An equilateral triangle has three sides of equal length.

EXAMPLE 12

The perimeter of an equilateral triangle is $93$ inches. Find the length of each side.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	Perimeter = 93 in.
Step 2. Identify what you are looking for.	length of the sides of an equilateral triangle
Step 3. Name. Choose a variable to represent it.	Let s = length of each side
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Each side is 31 inches

TRY IT 12.1

Find the length of each side of an equilateral triangle with perimeter $39$ inches.

Answer

13 in.

TRY IT 12.2

Find the length of each side of an equilateral triangle with perimeter $51$ centimetres.

Answer

17 cm

EXAMPLE 13

Arianna has $156$ inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of $60$ inches. How long can she make the two equal sides?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	P = 156 in.
Step 2. Identify what you are looking for.	the lengths of the two equal sides
Step 3. Name. Choose a variable to represent it.	Let s = the length of each side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Arianna can make each of the two equal sides 48 inches l

TRY IT 13.1

A backyard deck is in the shape of an isosceles triangle with a base of $20$ feet. The perimeter of the deck is $48$ feet. How long is each of the equal sides of the deck?

Answer

14 ft

TRY IT 13.2

A boat’s sail is an isosceles triangle with base of $8$ metres. The perimeter is $22$ metres. How long is each of the equal sides of the sail?

Answer

7 m

Use the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base $b$ , and the length of the bigger base $B$ . The height, $h$ , of a trapezoid is the distance between the two bases as shown in (Figure.13).

A trapezoid has a larger base, $B$ , and a smaller base, $b$ . The height $h$ is the distance between the bases.

A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.

Figure.13

Formula for the Area of a Trapezoid

${\text{Area}}_{\text{trapezoid}}=\frac{1}{2}h\left(b+B\right)$

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See (Figure.14).

Splitting a trapezoid into two triangles may help you understand the formula for its area.

An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.

Figure.14

The height of the trapezoid is also the height of each of the two triangles. See (Figure.15).

Figure.15

The formula for the area of a trapezoid is

This image shows the formula for the area of a trapezoid and says “area of trapezoid equals one-half h times smaller base b plus larger base B).

If we distribute, we get,

The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.

Properties of Trapezoids

A trapezoid has four sides. See (Figure.13).
Two of its sides are parallel and two sides are not.
The area, $A$ , of a trapezoid is $\text{A}=\frac{1}{2}h\left(b+B\right)$ .

EXAMPLE 14

Find the area of a trapezoid whose height is 6 inches and whose bases are $14$ and $11$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the trapezoid
Step 3. Name. Choose a variable to represent it.	Let $A=\text{the area}$
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable?

If we draw a rectangle around the trapezoid that has the same big base $B$ and a height $h$ , its area should be greater than that of the trapezoid.

If we draw a rectangle inside the trapezoid that has the same little base $b$ and a height $h$ , its area should be smaller than that of the trapezoid.

The area of the larger rectangle is $84$ square inches and the area of the smaller rectangle is $66$ square inches. So it makes sense that the area of the trapezoid is between $84$ and $66$ square inches

Step 7. Answer the question. The area of the trapezoid is $75$ square inches.

TRY IT 14.1

The height of a trapezoid is $14$ yards and the bases are $7$ and $16$ yards. What is the area?

Answer

161 sq. yd

TRY IT 14.2

The height of a trapezoid is $18$ centimetres and the bases are $17$ and $8$ centimetres. What is the area?

Answer

225 sq. cm

EXAMPLE 15

Find the area of a trapezoid whose height is $5$ feet and whose bases are $10.3$ and $13.7$ feet.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the trapezoid
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable? The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5.
Step 7. Answer the question.	The area of the trapezoid is 60 square feet.

TRY IT 15.1

The height of a trapezoid is $7$ centimetres and the bases are $4.6$ and $7.4$ centimetres. What is the area?

Answer

42 sq. cm

TRY IT 15.2

The height of a trapezoid is $9$ metres and the bases are $6.2$ and $7.8$ metres. What is the area?

Answer

63 sq. m

EXAMPLE 16

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of $3.4$ yards and the bases are $8.2$ and $5.6$ yards. How many square yards will be available to plant?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a trapezoid
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable? Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd.
Step 7. Answer the question.	Vinny has 23.46 square yards in which he can plan

TRY IT 16.1

Lin wants to sod his lawn, which is shaped like a trapezoid. The bases are $10.8$ yards and $6.7$ yards, and the height is $4.6$ yards. How many square yards of sod does he need?

Answer

40.25 sq. yd

TRY IT 16.2

Kira wants cover his patio with concrete pavers. If the patio is shaped like a trapezoid whose bases are $18$ feet and $14$ feet and whose height is $15$ feet, how many square feet of pavers will he need?

Answer

240 sq. ft.

Access Additional Online Resources

Key Concepts

Properties of Rectangles
- Rectangles have four sides and four right (90°) angles.
- The lengths of opposite sides are equal.
- The perimeter, , of a rectangle is the sum of twice the length and twice the width.
  - $P=2L+2W$
- The area, , of a rectangle is the length times the width.
  - $A=L\cdot W$
Triangle Properties
- For any triangle , the sum of the measures of the angles is 180°.
  - $m\angle A+m\angle B+m\angle C=180$ °
- The perimeter of a triangle is the sum of the lengths of the sides.
  - $P=a+b+c$
- The area of a triangle is one-half the base, b, times the height, h.
  - $A=\frac{1}{2}bh$

Glossary

area: The area is a measure of the surface covered by a figure.

equilateral triangle: A triangle with all three sides of equal length is called an equilateral triangle.

isosceles triangle: A triangle with two sides of equal length is called an isosceles triangle.

perimeter: The perimeter is a measure of the distance around a figure.

rectangle: A rectangle is a geometric figure that has four sides and four right angles.

trapezoid: A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not.

Practice Makes Perfect

Understand Linear, Square, and Cubic Measure

In the following exercises, determine whether you would measure each item using linear, square, or cubic units.

1. amount of water in a fish tank	2. length of dental floss
3. living area of an apartment	4. floor space of a bathroom tile
5. height of a doorway	6. capacity of a truck trailer

In the following exercises, find the a) perimeter and b) area of each figure. Assume each side of the square is $1$ cm.

7.	8.
9.	10.
11.	12.

Use the Properties of Rectangles

In the following exercises, find the a) perimeter and b) area of each rectangle.

13. The length of a rectangle is $85$ feet and the width is $45$ feet.	14. The length of a rectangle is $26$ inches and the width is $58$ inches.
15. A rectangular room is $15$ feet wide by $14$ feet long.	16. A driveway is in the shape of a rectangle $20$ feet wide by $35$ feet long.

In the following exercises, solve.

17. Find the length of a rectangle with perimeter $124$ inches and width $38$ inches.	18. Find the length of a rectangle with perimeter $20.2$ yards and width of $7.8$ yards.
19. Find the width of a rectangle with perimeter $92$ metres and length $19$ metres.	20. Find the width of a rectangle with perimeter $16.2$ metres and length $3.2$ metres.
21. The area of a rectangle is $414$ square metres. The length is $18$ metres. What is the width?	22. The area of a rectangle is $782$ square centimetres. The width is $17$ centimetres. What is the length?
23. The length of a rectangle is $9$ inches more than the width. The perimeter is $46$ inches. Find the length and the width.	24. The width of a rectangle is $8$ inches more than the length. The perimeter is $52$ inches. Find the length and the width.
25. The perimeter of a rectangle is $58$ metres. The width of the rectangle is $5$ metres less than the length. Find the length and the width of the rectangle.	26. The perimeter of a rectangle is $62$ feet. The width is $7$ feet less than the length. Find the length and the width.
27. The width of the rectangle is $0.7$ metres less than the length. The perimeter of a rectangle is $52.6$ metres. Find the dimensions of the rectangle.	28. The length of the rectangle is $1.1$ metres less than the width. The perimeter of a rectangle is $49.4$ metres. Find the dimensions of the rectangle.
29. The perimeter of a rectangle of $150$ feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.	30. The length of a rectangle is three times the width. The perimeter is $72$ feet. Find the length and width of the rectangle.
31. The length of a rectangle is $3$ metres less than twice the width. The perimeter is $36$ metres. Find the length and width.	32. The length of a rectangle is $5$ inches more than twice the width. The perimeter is $34$ inches. Find the length and width.
33. The width of a rectangular window is $24$ inches. The area is $624$ square inches. What is the length?	34. The length of a rectangular poster is $28$ inches. The area is $1316$ square inches. What is the width?
35. The area of a rectangular roof is $2310$ square metres. The length is $42$ metres. What is the width?	36. The area of a rectangular tarp is $132$ square feet. The width is $12$ feet. What is the length?
37. The perimeter of a rectangular courtyard is $160$ feet. The length is $10$ feet more than the width. Find the length and the width.	38. The perimeter of a rectangular painting is $306$ centimetres. The length is $17$ centimetres more than the width. Find the length and the width.
39. The width of a rectangular window is $40$ inches less than the height. The perimeter of the doorway is $224$ inches. Find the length and the width.	40. The width of a rectangular playground is $7$ metres less than the length. The perimeter of the playground is $46$ metres. Find the length and the width.

Use the Properties of Triangles

In the following exercises, solve using the properties of triangles.

41. Find the area of a triangle with base $12$ inches and height $5$ inches.	42. Find the area of a triangle with base $45$ centimetres and height $30$ centimetres.
43. Find the area of a triangle with base $8.3$ metres and height $6.1$ metres.	44. Find the area of a triangle with base $24.2$ feet and height $20.5$ feet.
45. A triangular flag has base of $1$ foot and height of $1.5$ feet. What is its area?	46. A triangular window has base of $8$ feet and height of $6$ feet. What is its area?
47. If a triangle has sides of $6$ feet and $9$ feet and the perimeter is $23$ feet, how long is the third side?	48. If a triangle has sides of $14$ centimetres and $18$ centimetres and the perimeter is $49$ centimetres, how long is the third side?
49. What is the base of a triangle with an area of $207$ square inches and height of $18$ inches?	50. What is the height of a triangle with an area of $893$ square inches and base of $38$ inches?
51. The perimeter of a triangular reflecting pool is $36$ yards. The lengths of two sides are $10$ yards and $15$ yards. How long is the third side?	52. A triangular courtyard has perimeter of $120$ metres. The lengths of two sides are $30$ metres and $50$ metres. How long is the third side?
53. An isosceles triangle has a base of $20$ centimetres. If the perimeter is $76$ centimetres, find the length of each of the other sides.	54. An isosceles triangle has a base of $25$ inches. If the perimeter is $95$ inches, find the length of each of the other sides.
55. Find the length of each side of an equilateral triangle with a perimeter of $51$ yards.	56. Find the length of each side of an equilateral triangle with a perimeter of $54$ metres.
57. The perimeter of an equilateral triangle is $18$ metres. Find the length of each side.	58. The perimeter of an equilateral triangle is $42$ miles. Find the length of each side.
59. The perimeter of an isosceles triangle is $42$ feet. The length of the shortest side is $12$ feet. Find the length of the other two sides.	60. The perimeter of an isosceles triangle is $83$ inches. The length of the shortest side is $24$ inches. Find the length of the other two sides.
61. A dish is in the shape of an equilateral triangle. Each side is $8$ inches long. Find the perimeter.	62. A floor tile is in the shape of an equilateral triangle. Each side is $1.5$ feet long. Find the perimeter.
63. A road sign in the shape of an isosceles triangle has a base of $36$ inches. If the perimeter is $91$ inches, find the length of each of the other sides.	64. A scarf in the shape of an isosceles triangle has a base of $0.75$ metres. If the perimeter is $2$ metres, find the length of each of the other sides.
65. The perimeter of a triangle is $39$ feet. One side of the triangle is $1$ foot longer than the second side. The third side is $2$ feet longer than the second side. Find the length of each side.	66. The perimeter of a triangle is $35$ feet. One side of the triangle is $5$ feet longer than the second side. The third side is $3$ feet longer than the second side. Find the length of each side.
67. One side of a triangle is twice the smallest side. The third side is $5$ feet more than the shortest side. The perimeter is $17$ feet. Find the lengths of all three sides.	68. One side of a triangle is three times the smallest side. The third side is $3$ feet more than the shortest side. The perimeter is $13$ feet. Find the lengths of all three sides.

Use the Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

69. The height of a trapezoid is $12$ feet and the bases are $9$ and $15$ feet. What is the area?	70. The height of a trapezoid is $24$ yards and the bases are $18$ and $30$ yards. What is the area?
71. Find the area of a trapezoid with a height of $51$ metres and bases of $43$ and $67$ metres.	72. Find the area of a trapezoid with a height of $62$ inches and bases of $58$ and $75$ inches.
73. The height of a trapezoid is $15$ centimetres and the bases are $12.5$ and $18.3$ centimetres. What is the area?	74. The height of a trapezoid is $48$ feet and the bases are $38.6$ and $60.2$ feet. What is the area?
75. Find the area of a trapezoid with a height of $4.2$ metres and bases of $8.1$ and $5.5$ metres.	76. Find the area of a trapezoid with a height of $32.5$ centimetres and bases of $54.6$ and $41.4$ centimetres.
77. Laurel is making a banner shaped like a trapezoid. The height of the banner is $3$ feet and the bases are $4$ and $5$ feet. What is the area of the banner?	78. Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width $5$ feet and lengths $5$ feet and $8$ feet. What is the area of the floor?
79. Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width $18.5$ inches and lengths $62$ and $50$ inches. What is the area of the counter?	80. Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width $8$ inches and lengths $48.2$ inches and $56.2$ inches. What is the area of the scarf?

Everyday Math

81. Fence Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a $50$ foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be $10$ feet. How long can he make the other side if he wants to use the entire roll of fence?

82. Gardening Lupita wants to fence in her tomato garden. The garden is rectangular and the length is twice the width. It will take $48$ feet of fencing to enclose the garden. Find the length and width of her garden.

83. Fence Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are $6$ feet, $8$ feet, and $10$ feet. The fence costs $\$10$ per foot. How much will it cost for Christa to fence in her flowerbed?

84. Painting Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with height $8$ feet and bases $20$ feet and $12$ feet. The cost of the painting one square foot of wall is about $?0.05$ . About how much will it cost for Caleb to paint the attic wall?

A right trapezoid is shown.

Writing Exercises

86. If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.

87. Look at the two figures.

A rectangle is shown on the left. It is labeled as 2 by 8. A square is shown on the right. It is labeled as 4 by 4.

a) Which figure looks like it has the larger area? Which looks like it has the larger perimeter?

b) Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?

88. The length of a rectangle is $5$ feet more than the width. The area is $50$ square feet. Find the length and the width.

a) Write the equation you would use to solve the problem.

b) Why can’t you solve this equation with the methods you learned in the previous chapter?

Answers

1. cubic	3. square	5. linear
7. a) 10 cm, b) 4 sq. cm	9. a) 8 cm, b) 3 sq. cm	11. a) 10 cm, b) 5 sq. cm
13. a) 260 ft, b) 3825 sq. ft	15. a) 58 ft, b) 210 sq. ft	17. 24 inches
19. 27 metres	21. 23 m	23. 7 in., 16 in.
25. 17 m, 12 m	27. 13.5 m, 12.8 m	29. 25 ft, 50 ft
31. 7 m, 11 m	33. 26 in.	35. 55 m
37. 35 ft, 45 ft	39. 76 in., 36 in.	41. 60 sq. in.
43. 25.315 sq. m	45. 0.75 sq. ft	47. 8 ft
49. 23 in.	51. 11 ft	53. 28 cm
55. 17 ft	57. 6 m	59. 15 ft
61. 24 in.	63. 27.5 in.	65. 12 ft, 13 ft, 14 ft
67. 3 ft, 6 ft, 8 ft	69. 144 sq. ft	71. 2805 sq. m
73. 231 sq. cm	75. 28.56 sq. m	77. 13.5 sq. ft
79. 1036 sq. in.	81. 15 ft	83. $24
85. Answers will vary.	87. Answers will vary.

Attributions

This chapter has been adapted from “Use Properties of Rectangles, Triangles, and Trapezoids” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

4.3 Solve Geometry Applications: Volume and Surface Area

Learning Objectives

By the end of this section, you will be able to:

Find volume and surface area of rectangular solids
Find volume and surface area of spheres
Find volume and surface area of cylinders
Find volume of cone

In this section, we will find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

Problem Solving Strategy for Geometry Applications

Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for. Choose a variable to represent that quantity.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

Find Volume and Surface Area of Rectangular Solids

A cheer leading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See Figure.1). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

This wooden crate is in the shape of a rectangular solid.

Figure.1

Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in Figure.2 has length $4$ units, width $2$ units, and height $3$ units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This $4$ by $2$ by $3$ rectangular solid has $24$ cubic units.

A rectangular solid is shown. Each layer is composed of 8 cubes, measuring 2 by 4. The top layer is pink. The middle layer is orange. The bottom layer is green. Beside this is an image of the top layer that says “The top layer has 8 cubic units.” The orange layer is shown and says “The middle layer has 8 cubic units.” The green layer is shown and says, “The bottom layer has 8 cubic units.”

Figure.2

Altogether there are $24$ cubic units. Notice that $24$ is the $\text{length}\phantom{\rule{1.0em}{0ex}}\times\phantom{\rule{1.0em}{0ex}}\text{width}\phantom{\rule{1.0em}{0ex}}\times\phantom{\rule{1.0em}{0ex}}\text{height}\text{.}$

The top line says V equals L times W times H. Beneath the V is 24, beneath the equal sign is another equal sign, beneath the L is a 4, beneath the W is a 2, beneath the H is a 3.

The volume, $V$ , of any rectangular solid is the product of the length, width, and height.

$V=LWH$

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, $B$ , is equal to $\text{length}\times\text{width}\text{.}$

$\text{B}=\text{L}\cdot\text{W}$

We can substitute $\text{B}$ for $\text{L}\cdot\text{W}$ in the volume formula to get another form of the volume formula.

The top line says V equals red L times red W times H. Below this is V equals red parentheses L times W times H. Below this is V equals red capital B times h.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the $4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}3$ rectangular solid we started with. See Figure.3.

An image of a rectangular solid is shown. It is made up of cubes. It is labeled as 2 by 4 by 3. Beside the solid is V equals Bh. Below this is V equals Base times height. Below Base is parentheses 4 times 2. The next line says V equals parentheses 4 times 2 times 3. Below that is V equals 8 times 3, then V equals 24 cubic units.

Figure.3

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

$\begin{array}{ccccccc}{A}_{\text{front}}=L\times W\hfill & & & {A}_{\text{side}}=L\times W\hfill & & & {A}_{\text{top}}=L\times W\hfill \\ {A}_{\text{front}}=4\cdot3\hfill & & & {A}_{\text{side}}=2\cdot3\hfill & & & {A}_{\text{top}}=4\cdot2\hfill \\ {A}_{\text{front}}=12\hfill & & & {A}_{\text{side}}=6\hfill & & & {A}_{\text{top}}=8\hfill \end{array}$

Notice for each of the three faces you see, there is an identical opposite face that does not show.

$\begin{array}{l}S=\left(\text{front}+\text{back}\right)\text{+}\left(\text{left side}+\text{right side}\right)+\left(\text{top}+\text{bottom}\right)\\ S=\left(2\cdot\text{front}\right)+\left(\text{2}\cdot\text{left side}\right)+\left(\text{2}\cdot\text{top}\right)\\ S=2\cdot12+2\cdot6+2\cdot8\\ S=24+12+16\\ S=52\phantom{\rule{0.2em}{0ex}}\text{sq. units}\end{array}$

The surface area $S$ of the rectangular solid shown in (Figure.3) is $52$ square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see Figure.4). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

$S=2LH+2LW+2WH$

For each face of the rectangular solid facing you, there is another face on the opposite side. There are $6$ faces in all.

A rectangular solid is shown. The sides are labeled L, W, and H. One face is labeled LW and another is labeled WH.

Figure.4

Volume and Surface Area of a Rectangular Solid

For a rectangular solid with length $L$ , width $W$ , and height $H:$

A rectangular solid is shown. The sides are labeled L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.

EXAMPLE 1

For a rectangular solid with length $14$ cm, height $17$ cm, and width $9$ cm, find the a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and
label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the rectangular solid
Step 3. Name. Choose a variable to represent it.	Let $V$ = volume
Step 4. Translate. Write the appropriate formula. Substitute.	$V=LWH$ $V=\mathrm{14}\cdot 9\cdot 17$
Step 5. Solve the equation.	$V=2,142$
Step 6. Check We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is $\text{1,034}$ square centimetres.

b)
Step 2. Identify what you are looking for.	the surface area of the solid
Step 3. Name. Choose a variable to represent it.	Let $S$ = surface area
Step 4. Translate. Write the appropriate formula. Substitute.	$S=2LH+2LW+2WH$ $S=2\left(14\cdot 17\right)+2\left(14\cdot 9\right)+2\left(9\cdot 17\right)$
Step 5. Solve the equation.	$S=1,034$
Step 6. Check: Double-check with a calculator.
Step 7. Answer the question.	The surface area is 1,034 square centimetres.

TRY IT 1.1

Find the a) volume and b) surface area of rectangular solid with the: length $8$ feet, width $9$ feet, and height $11$ feet.

Answer

792 cu. ft
518 sq. ft

TRY IT 1.2

Find the a) volume and b) surface area of rectangular solid with the: length $15$ feet, width $12$ feet, and height $8$ feet.

Answer

1,440 cu. ft
792 sq. ft

EXAMPLE 2

A rectangular crate has a length of $30$ inches, width of $25$ inches, and height of $20$ inches. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and
label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the crate
Step 3. Name. Choose a variable to represent it.	let $V$ = volume
Step 4. Translate. Write the appropriate formula. Substitute.	$V=LWH$ $V=30\cdot 25\cdot 20$
Step 5. Solve the equation.	$V=15,000$
Step 6. Check: Double check your math.
Step 7. Answer the question.	The volume is 15,000 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the crate
Step 3. Name. Choose a variable to represent it.	let $S$ = surface area
Step 4. Translate. Write the appropriate formula. Substitute.	$S=2LH+2LW+2WH$ $S=2\left(30\cdot 20\right)+2\left(30\cdot 25\right)+2\left(25\cdot 20\right)$
Step 5. Solve the equation.	$S=3,700$
Step 6. Check: Check it yourself!
Step 7. Answer the question.	The surface area is 3,700 square inches.

TRY IT 2.1

A rectangular box has length $9$ feet, width $4$ feet, and height $6$ feet. Find its a) volume and b) surface area.

Answer

216 cu. ft
228 sq. ft

TRY IT 2.2

A rectangular suitcase has length $22$ inches, width $14$ inches, and height $9$ inches. Find its a) volume and b) surface area.

Answer

2,772 cu. in.
1,264 sq. in.

Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

$\begin{array}{ccccc}V=LWH\hfill & & & & S=2LH+2LW+2WH\hfill \\ V=\text{s}\cdot\text{s}\cdot\text{s}\hfill & & & & S=2\text{s}\cdot\text{s}+2\text{s}\cdot\text{s}+2\text{s}\cdot\text{s}\hfill \\ V={\text{s}}^{3}\hfill & & & & S=2{s}^{2}+2{s}^{2}+2{s}^{2}\hfill \\ & & & & S=6{s}^{2}\hfill \end{array}$

So for a cube, the formulas for volume and surface area are $V={s}^{3}$ and $S=6{s}^{2}$ .

Volume and Surface Area of a Cube

For any cube with sides of length $s$ ,

An image of a cube is shown. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.

EXAMPLE 3

A cube is $2.5$ inches on each side. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and
label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cube
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula.	$V={s}^{3}$
Step 5. Solve. Substitute and solve.	$V={\left(2.5\right)}^{3}$ $V=15.625$
Step 6. Check: Check your work.
Step 7. Answer the question.	The volume is 15.625 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.	$S=6{s}^{2}$
Step 5. Solve. Substitute and solve.	$S=6\cdot {\left(2.5\right)}^{2}$ $S=37.5$
Step 6. Check: The check is left to you.
Step 7. Answer the question.	The surface area is 37.5 square inches.

TRY IT 3.1

For a cube with side 4.5 metres, find the a) volume and b) surface area of the cube.

Answer

91.125 cu. m
121.5 sq. m

TRY IT 3.2

For a cube with side 7.3 yards, find the a) volume and b) surface area of the cube.

Answer

389.017 cu. yd.
319.74 sq. yd.

EXAMPLE 4

A notepad cube measures $2$ inches on each side. Find its a) volume and b) surface area.

Solution

Step 1. Read the problem. Draw the figure and
label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cube
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula.	$V={s}^{3}$
Step 5. Solve the equation.	$V={2}^{3}$ $V=8$
Step 6. Check: Check that you did the calculations correctly.
Step 7. Answer the question.	The volume is 8 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.	$S=6{s}^{2}$
Step 5. Solve the equation.	$S=6\cdot {2}^{2}$ $S=24$
Step 6. Check: The check is left to you.
Step 7. Answer the question.	The surface area is 24 square inches.

TRY IT 4.1

A packing box is a cube measuring $4$ feet on each side. Find its a) volume and b) surface area.

Answer

64 cu. ft
96 sq. ft

TRY IT 4.2

An unopened tissue box is a cube measuring $5$ inch on each side. Find its a) volume and b) surface area.

Answer

125 cu. in
150 sq. in

Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the centre of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate $\pi$ with $3.14$ .

Volume and Surface Area of a Sphere

For a sphere with radius $r\text{:}$

An image of a sphere is shown. The radius is labeled r. Beside this is Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.

EXAMPLE 5

A sphere has a radius $6$ inches. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and label
it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the sphere
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula.	$V=\frac{4}{3}\pi {r}^{3}$
Step 5. Solve.	$V\approx \frac{4}{3}\left(3.14\right){6}^{3}$ $V\approx 904.32\phantom{\rule{0.2em}{0ex}}\text{cubic inches}$
Step 6. Check: Double-check your math on a calculator.
Step 7. Answer the question.	The volume is approximately 904.32 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.	$S=4\pi {r}^{2}$
Step 5. Solve.	$S\approx 4\left(3.14\right){6}^{2}$ $S\approx 452.16$
Step 6. Check: Double-check your math on a calculator
Step 7. Answer the question.	The surface area is approximately 452.16 square inches.

TRY IT 5.1

Find the a) volume and b) surface area of a sphere with radius 3 centimetres.

Answer

113.04 cu. cm
113.04 sq. cm

TRY IT 5.2

Find the a) volume and b) surface area of each sphere with a radius of $1$ foot

Answer

4.19 cu. ft
12.56 sq. ft

EXAMPLE 6

A globe of Earth is in the shape of a sphere with radius $14$ centimetres. Find its a) volume and b) surface area. Round the answer to the nearest hundredth.

Solution

Step 1. Read the problem. Draw a figure with the
given information and label it.

a)
Step 2. Identify what you are looking for.	the volume of the sphere
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$V=\frac{4}{3}\pi {r}^{3}$ $V\approx \frac{4}{3}\left(3.14\right){14}^{3}$
Step 5. Solve.	$V\approx 11,488.21$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 11,488.21 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the sphere
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$S=4\pi {r}^{2}$ $S\approx 4\left(3.14\right){14}^{2}$
Step 5. Solve.	$S\approx 2461.76$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately 2461.76 square inches.

TRY IT 6.1

A beach ball is in the shape of a sphere with radius of $9$ inches. Find its a) volume and b) surface area.

Answer

3052.08 cu. in.
1017.36 sq. in.

TRY IT 6.2

A Roman statue depicts Atlas holding a globe with radius of $1.5$ feet. Find the a) volume and b) surface area of the globe.

Answer

14.13 cu. ft
28.26 sq. ft

Find the Volume and Surface Area of a Cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height $h$ of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, $h$ , will be perpendicular to the bases.

A cylinder has two circular bases of equal size. The height is the distance between the bases.

An image of a cylinder is shown. There is a red arrow pointing to the radius of the top labeling it r, radius. There is a red arrow pointing to the height of the cylinder labeling it h, height.

Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, $V=Bh$ , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, $B$ , is the area of the rectangular base, length × width. For a cylinder, the area of the base, $B$ , is the area of its circular base, $\pi {r}^{2}$ . (Figure.5) compares how the formula $V=Bh$ is used for rectangular solids and cylinders.

Seeing how a cylinder is similar to a rectangular solid may make it easier to understand the formula for the volume of a cylinder.

In (a), a rectangular solid is shown. The sides are labeled L, W, and H. Below this is V equals capital Bh, then V equals Base times h, then V equals parentheses lw times h, then V equals lwh. In (b), a cylinder is shown. The radius of the top is labeled r, the height is labeled h. Below this is V equals capital Bh, then V equals Base times h, then V equals parentheses pi r squared times h, then V equals pi times r squared times h.

Figure.5

To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See (Figure.6).

By cutting and unrolling the label of a can of vegetables, we can see that the surface of a cylinder is a rectangle. The length of the rectangle is the circumference of the cylinder’s base, and the width is the height of the cylinder.

A cylindrical can of green beans is shown. The height is labeled h. Beside this are pictures of circles for the top and bottom of the can and a rectangle for the other portion of the can. Above the circles is C equals 2 times pi times r. The top of the rectangle says l equals 2 times pi times r. The left side of the rectangle is labeled h, the right side is labeled w.

Figure.6

The distance around the edge of the can is the circumference of the cylinder’s base it is also the length $L$ of the rectangular label. The height of the cylinder is the width $W$ of the rectangular label. So the area of the label can be represented as

The top line says A equals l times red w. Below the l is 2 times pi times r. Below the w is a red h.

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

A rectangle is shown with circles coming off the top and bottom.

The surface area of a cylinder with radius $r$ and height $h$ , is

$S=2\pi {r}^{2}+2\pi rh$

Volume and Surface Area of a Cylinder

For a cylinder with radius $r$ and height $h:$

A cylinder is shown. The height is labeled h and the radius of the top is labeled r. Beside it is Volume: V equals pi times r squared times h or V equals capital B times h. Below this is Surface Area: S equals 2 times pi times r squared plus 2 times pi times r times h.

EXAMPLE 7

A cylinder has height $5$ centimetres and radius $3$ centimetres. Find the a) volume and b) surface area.

Solution

Step 1. Read the problem. Draw the figure and label
it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cylinder
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$V=\pi {r}^{2}h$ $V\approx \left(3.14\right){3}^{2}\cdot 5$
Step 5. Solve.	$V\approx 141.3$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 141.3 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cylinder
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$S=2\pi {r}^{2}+2\pi rh$ $S\approx 2\left(3.14\right){3}^{2}+2\left(3.14\right)\left(3\right)5$
Step 5. Solve.	$S\approx 150.72$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately 150.72 square inches.

TRY IT 7.1

Find the a) volume and b) surface area of the cylinder with radius 4 cm and height 7cm.

Answer

351.68 cu. cm
276.32 sq. cm

TRY IT 7.2

Find the a) volume and b) surface area of the cylinder with given radius 2 ft and height 8 ft.

Answer

100.48 cu. ft
125.6 sq. ft

EXAMPLE 8

Find the a) volume and b) surface area of a can of soda. The radius of the base is $4$ centimetres and the height is $13$ centimetres. Assume the can is shaped exactly like a cylinder.

Solution

Step 1. Read the problem. Draw the figure and
label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cylinder
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$V=\pi {r}^{2}h$ $V\approx \left(3.14\right){4}^{2}\cdot 13$
Step 5. Solve.	$V\approx 653.12$
Step 6. Check: We leave it to you to check.
Step 7. Answer the question.	The volume is approximately 653.12 cubic centimetres.

b)
Step 2. Identify what you are looking for.	the surface area of the cylinder
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$S=2\pi {r}^{2}+2\pi rh$ $S\approx 2\left(3.14\right){4}^{2}+2\left(3.14\right)\left(4\right)13$
Step 5. Solve.	$S\approx 427.04$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately 427.04 square centimetres.

TRY IT 8.1

Find the a) volume and b) surface area of a can of paint with radius 8 centimetres and height 19 centimetres. Assume the can is shaped exactly like a cylinder.

Answer

3,818.24 cu. cm
1,356.48 sq. cm

TRY IT 8.2

Find the a) volume and b) surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.

Answer

91.5624 cu. ft
113.6052 sq. ft

Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See (Figure.6).

The height of a cone is the distance between its base and the vertex.

An image of a cone is shown. The top is labeled vertex. The height is labeled h. The radius of the base is labeled r.

Figure.6

Earlier in this section, we saw that the volume of a cylinder is $V=\pi{r}^{2}h$ . We can think of a cone as part of a cylinder. Figure.7 shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.

The volume of a cone is less than the volume of a cylinder with the same base and height.

An image of a cone is shown. There is a cylinder drawn around it.

Figure.7

In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is

The formula V equals one-third times capital B times h is shown.

Since the base of a cone is a circle, we can substitute the formula of area of a circle, $\pi{r}^{2}$ , for $B$ to get the formula for volume of a cone.

The formula V equals one-third times pi times r squared times h is shown.

In this book, we will only find the volume of a cone, and not its surface area.

Volume of a Cone

For a cone with radius $r$ and height $h$ .

An image of a cone is shown. The height is labeled h, the radius of the base is labeled r. Beside this is Volume: V equals one-third times pi times r squared times h.

EXAMPLE 9

Find the volume of a cone with height $6$ inches and radius of its base $2$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the volume of the cone
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$V=\frac{1}{3}\phantom{\rule{1em}{0ex}}\pi \phantom{\rule{1.9em}{0ex}}{r}^{2}\phantom{\rule{1.7em}{0ex}}h$ $V\approx \frac{1}{3}\phantom{\rule{0.7em}{0ex}}3.14\phantom{\rule{1em}{0ex}}{\left(2\right)}^{2}\phantom{\rule{1em}{0ex}}\left(6\right)$
Step 5. Solve.	$V\approx 25.12$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 25.12 cubic inches.

TRY IT 9.1

Find the volume of a cone with height $7$ inches and radius $3$ inches

Answer

65.94 cu. in.

TRY IT 9.2

Find the volume of a cone with height $9$ centimetres and radius $5$ centimetres

Answer

235.5 cu. cm

EXAMPLE 10

Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is $8$ inches tall and $5$ inches in diametre? Round the answer to the nearest hundredth.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.
Step 2. Identify what you are looking for.	the volume of the cone
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ , and notice that we were given the distance across the circle, which is its diametre. The radius is 2.5 inches.)	$V=\frac{1}{3}\phantom{\rule{1em}{0ex}}\pi \phantom{\rule{2.3em}{0ex}}{r}^{2}\phantom{\rule{2em}{0ex}}h$ $V\approx \frac{1}{3}\phantom{\rule{0.7em}{0ex}}3.14\phantom{\rule{1em}{0ex}}{\left(2.5\right)}^{2}\phantom{\rule{1em}{0ex}}\left(8\right)$
Step 5. Solve.	$V\approx 52.33$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume of the wrap is approximately 52.33 cubic inches.

TRY IT 10.1

How many cubic inches of candy will fit in a cone-shaped piñata that is $18$ inches long and $12$ inches across its base? Round the answer to the nearest hundredth.

Answer

678.24 cu. in.

TRY IT 10.2

What is the volume of a cone-shaped party hat that is $10$ inches tall and $7$ inches across at the base? Round the answer to the nearest hundredth.

Answer

128.2 cu. in.

ACCESS ADDITIONAL ONLINE RESOURCES

Volume of a Cone

Key Concepts

Volume and Surface Area of a Rectangular Solid
- $V=LWH$
- $S=2LH+2LW+2WH$
Volume and Surface Area of a Cube
- $V={s}^{3}$
- $S=6{s}^{2}$
Volume and Surface Area of a Sphere
- $V=\frac{4}{3}\pi {r}^{3}$
- $S=4\pi {r}^{2}$
Volume and Surface Area of a Cylinder
- $V=\pi {r}^{2}h$
- $S=2\pi {r}^{2}+2\pi rh$
Volume of a Cone
- For a cone with radius $r$ and height $h$ :
  Volume: $V=\frac{1}{3}\pi {r}^{2}h$

Glossary

cone: A cone is a solid figure with one circular base and a vertex.

cube: A cube is a rectangular solid whose length, width, and height are equal.

cylinder: A cylinder is a solid figure with two parallel circles of the same size at the top and bottom.

Practice Makes Perfect

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find a) the volume and b) the surface area of the rectangular solid with the given dimensions.

1. length $2$ metres, width $1.5$ metres, height $3$ metres	2. length $5$ feet, width $8$ feet, height $2.5$ feet
3. length $3.5$ yards, width $2.1$ yards, height $2.4$ yards	4. length $8.8$ centimetres, width $6.5$ centimetres, height $4.2$ centimetres

In the following exercises, solve.

5. Moving van A rectangular moving van has length $16$ feet, width $8$ feet, and height $8$ feet. Find its a) volume and b) surface area.	6. Gift box A rectangular gift box has length $26$ inches, width $16$ inches, and height $4$ inches. Find its a) volume and b) surface area.
7. Carton A rectangular carton has length $21.3$ cm, width $24.2$ cm, and height $6.5$ cm. Find its a) volume and b) surface area.	8.Shipping container A rectangular shipping container has length $22.8$ feet, width $8.5$ feet, and height $8.2$ feet. Find its a) volume and b) surface area.

In the following exercises, find a) the volume and b) the surface area of the cube with the given side length.

9. $5$ centimetres	10. $6$ inches
11. $10.4$ feet	12. $12.5$ metres

In the following exercises, solve.

13. Science center Each side of the cube at the Discovery Science Center in Santa Ana is $64$ feet long. Find its a) volume and b) surface area.	14. Museum A cube-shaped museum has sides $45$ metres long. Find its a) volume and b) surface area.
15. Base of statue The base of a statue is a cube with sides $2.8$ metres long. Find its a) volume and b) surface area.	16. Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its a) volume and b) surface area.

Find the Volume and Surface Area of Spheres

In the following exercises, find a) the volume and b) the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

17. $3$ centimetres	18. $9$ inches
19. $7.5$ feet	20. $2.1$ yards

In the following exercises, solve. Round answers to the nearest hundredth.

21. Exercise ball An exercise ball has a radius of $15$ inches. Find its a) volume and b) surface area.	22. Balloon ride The Great Park Balloon is a big orange sphere with a radius of $36$ feet . Find its a) volume and b) surface area.
23. Golf ball A golf ball has a radius of $4.5$ centimetres. Find its a) volume and b) surface area.	24. Baseball A baseball has a radius of $2.9$ inches. Find its a) volume and b) surface area.

Find the Volume and Surface Area of a Cylinder

In the following exercises, find a) the volume and b) the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.

25. radius $3$ feet, height $9$ feet	26. radius $5$ centimetres, height $15$ centimetres
27. radius $1.5$ metres, height $4.2$ metres	28. radius $1.3$ yards, height $2.8$ yards

In the following exercises, solve. Round answers to the nearest hundredth.

29. Coffee can A can of coffee has a radius of $5$ cm and a height of $13$ cm. Find its a) volume and b) surface area.	30. Snack pack A snack pack of cookies is shaped like a cylinder with radius $4$ cm and height $3$ cm. Find its a) volume and b) surface area.
31. Barber shop pole A cylindrical barber shop pole has a diametre of $6$ inches and height of $24$ inches. Find its a) volume and b) surface area.	32. Architecture A cylindrical column has a diametre of $8$ feet and a height of $28$ feet. Find its a) volume and b) surface area.

Find the Volume of Cones

In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.

33. height $9$ feet and radius $2$ feet	34. height $8$ inches and radius $6$ inches
35. height $12.4$ centimetres and radius $5$ cm	36. height $15.2$ metres and radius $4$ metres

In the following exercises, solve. Round answers to the nearest hundredth.

37. Teepee What is the volume of a cone-shaped teepee tent that is $10$ feet tall and $10$ feet across at the base?	38. Popcorn cup What is the volume of a cone-shaped popcorn cup that is $8$ inches tall and $6$ inches across at the base?
39. Silo What is the volume of a cone-shaped silo that is $50$ feet tall and $70$ feet across at the base?	40. Sand pile What is the volume of a cone-shaped pile of sand that is $12$ metres tall and $30$ metres across at the base?

Everyday Math

41. Street light post The post of a street light is shaped like a truncated cone, as shown in the picture below. It is a large cone minus a smaller top cone. The large cone is $30$ feet tall with base radius $1$ foot. The smaller cone is $10$ feet tall with base radius of $0.5$ feet. To the nearest tenth,

a) find the volume of the large cone.

b) find the volume of the small cone.

c) find the volume of the post by subtracting the volume of the small cone from the volume of the large cone.

An image of a cone is shown. There is a dark dotted line at the top indicating a smaller cone.

42. Ice cream cones A regular ice cream cone is 4 inches tall and has a diametre of $2.5$ inches. A waffle cone is $7$ inches tall and has a diametre of $3.25$ inches. To the nearest hundredth,

a) find the volume of the regular ice cream cone.

b) find the volume of the waffle cone.

c) how much more ice cream fits in the waffle cone compared to the regular cone?

Writing Exercises

43. The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape.

44. Which has a larger volume, a cube of sides of $8$ feet or a sphere with a diametre of $8$ feet? Explain your reasoning.

Answers

a) 9 cu. m

b) 27 sq. m

a) 17.64 cu. yd.

b) 41.58 sq. yd.

a) 1,024 cu. ft

b) 640 sq. ft

a) 3,350.49 cu. cm

b) 1,622.42 sq. cm

a) 125 cu. cm

b) 150 sq. cm

11.

a) 1124.864 cu. ft.

b) 648.96 sq. ft

13.

a) 262,144 cu. ft

b) 24,576 sq. ft

15.

a) 21.952 cu. m

b) 47.04 sq. m

17.

a) 113.04 cu. cm

b) 113.04 sq. cm

19.

a) 1,766.25 cu. ft

b) 706.5 sq. ft

21.

a) 14,130 cu. in.

b) 2,826 sq. in.

23.

a) 381.51 cu. cm

b) 254.34 sq. cm

25.

a) 254.34 cu. ft

b) 226.08 sq. ft

27.

a) 29.673 cu. m

b) 53.694 sq. m

29.

a) 1,020.5 cu. cm

b) 565.2 sq. cm

31.

a) 678.24 cu. in.

b) 508.68 sq. in.

33. 37.68 cu. ft

35. 324.47 cu. cm

37. 261.67 cu. ft

39. 64,108.33 cu. ft

41.

a) 31.4 cu. ft

b) 2.6 cu. ft

c) 28.8 cu. ft

43. Answers will vary.

Attributions

This chapter has been adapted from “Solve Geometry Applications: Volume and Surface Area” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

4.4 Solve Geometry Applications: Circles and Irregular Figures

Learning Objectives

By the end of this section, you will be able to:

Use the properties of circles
Find the area of irregular figures

In this section, we’ll continue working with geometry applications. We will add several new formulas to our collection of formulas. To help you as you do the examples and exercises in this section, we will show the Problem Solving Strategy for Geometry Applications here.

Problem Solving Strategy for Geometry Applications

Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for. Choose a variable to represent that quantity.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

Use the Properties of Circles

We’ll refer to the properties of circles as we use them to solve applications.

Properties of Circles

An image of a circle is shown. There is a line drawn through the widest part at the centre of the circle with a red dot indicating the centre of the circle. The line is labeled d. The two segments from the centre of the circle to the outside of the circle are each labeled r.

$r$ is the length of the radius
$d$ is the length of the diametre
$d=2r$
Circumference is the perimeter of a circle. The formula for circumference is
$C=2\pi r$
The formula for area of a circle is
$A=\pi {r}^{2}$

Remember, that we approximate $\pi$ with $3.14$ or $\frac{22}{7}$ depending on whether the radius of the circle is given as a decimal or a fraction. If you use the $\pi$ key on your calculator to do the calculations in this section, your answers will be slightly different from the answers shown. That is because the $\pi$ key uses more than two decimal places.

EXAMPLE 1

A circular sandbox has a radius of $2.5$ feet. Find the a) circumference and b) area of the sandbox.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the circle
Step 3. Name. Choose a variable to represent it.	Let c = circumference of the circle
Step 4. Translate. Write the appropriate formula Substitute	$C=2\pi r$ $C=2\pi \left(2.5\right)$
Step 5. Solve the equation.	$C\approx 2\left(3.14\right)\left(2.5\right)$ $C\approx 15\text{ft}$
Step 6. Check. Does this answer make sense? Yes. If we draw a square around the circle, its sides would be 5 ft (twice the radius), so its perimeter would be 20 ft. This is slightly more than the circle’s circumference, 15.7 ft.
Step 7. Answer the question.	The circumference of the sandbox is 15.7 feet.

Step 1. Read the problem. Draw the figure and label it with the given information
Step 2. Identify what you are looking for.	the area of the circle
Step 3. Name. Choose a variable to represent it.	Let A = the area of the circle
Step 4. Translate. Write the appropriate formula Substitute	$A=\pi{r}^{2}$ $A=\pi{\left(2.5\right)}^{2}$
Step 5. Solve the equation.	$A\approx \left(3.14\right){\left(2.5\right)}^{2}$ $A\approx 19.625\phantom{\rule{0.2em}{0ex}}\text{sq. ft}$
Step 6. Check. Yes. If we draw a square around the circle, its sides would be 5 ft, as shown in part a). So the area of the square would be 25 sq. ft. This is slightly more than the circle’s area, 19.625 sq. ft.
Step 7. Answer the question.	The area of the circle is 19.625 square feet.

TRY IT 1.1

A circular mirror has radius of $5$ inches. Find the a) circumference and b) area of the mirror.

Answer

31.4 in.
78.5 sq. in.

TRY IT 1.2

A circular spa has radius of $4.5$ feet. Find the a) circumference and b) area of the spa.

Answer

28.26 ft
63.585 sq. ft

We usually see the formula for circumference in terms of the radius $r$ of the circle:

$C=2\pi r$

But since the diametre of a circle is two times the radius, we could write the formula for the circumference in terms $\text{of}\phantom{\rule{0.2em}{0ex}}d$ .

$\begin{array}{cccc}& & & C=2\pi r\hfill \\ \text{Using the commutative property, we get}\hfill & & & C=\pi \cdot2r\hfill \\ \text{Then substituting}\phantom{\rule{0.2em}{0ex}}d=2r\hfill & & & C=\pi \cdot\text{d}\hfill \\ \text{So}\hfill & & & C=\pi d\hfill \end{array}$

We will use this form of the circumference when we’re given the length of the diametre instead of the radius.

EXAMPLE 2

A circular table has a diametre of four feet. What is the circumference of the table?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the table
Step 3. Name. Choose a variable to represent it.	Let c = the circumference of the table
Step 4. Translate. Write the appropriate formula for the situation. Substitute.	$C=\pi d$ $C=\pi \left(4\right)$
Step 5. Solve the equation, using 3.14 for $\pi$ .	$C\approx \left(3.14\right)\left(4\right)$ $C\approx 12.56\phantom{\rule{0.2em}{0ex}}\text{feet}$
Step 6. Check: If we put a square around the circle, its side would be 4. The perimeter would be 16. It makes sense that the circumference of the circle, 12.56, is a little less than 16.
Step 7. Answer the question.	The diametre of the table is 12.56 square feet

TRY IT 2.1

Find the circumference of a circular fire pit whose diametre is $5.5$ feet.

Answer

17.27 ft

TRY IT 2.2

If the diametre of a circular trampoline is $12$ feet, what is its circumference?

Answer

37.68 ft

EXAMPLE 3

Find the diametre of a circle with a circumference of $47.1$ centimetres.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the diametre of the circle
Step 3. Name. Choose a variable to represent it.	Let d = the diametre of the circle
Step 4. Translate.
Write the formula. Substitute, using 3.14 to approximate $\pi$ .
Step 5. Solve.
Step 6. Check: $47.1\stackrel{?}{=}\left(3.14\right)\left(15\right)$ $47.1=47.1$ ✓
Step 7. Answer the question.	The diametre of the circle is approximately 15 centimetres.

TRY IT 3.1

Find the diametre of a circle with circumference of $94.2$ centimetres.

Answer

30 cm

TRY IT 3.2

Find the diametre of a circle with circumference of $345.4$ feet.

Answer

110 ft

Find the Area of Irregular Figures

So far, we have found area for rectangles, triangles, trapezoids, and circles. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

EXAMPLE 4

Find the area of the shaded region.

Solution

The given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.

An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.

The blue rectangle has a width of $12$ and a length of $4$ . The red rectangle has a width of $2$ , but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is $4$ units long, the length of the red rectangle must be $6$ units.

An image of a blue horizontal rectangle attached to a red vertical rectangle is shown. The top is labeled 12, the side of the blue rectangle is labeled 4. The whole side is labeled 10, the blue portion is labeled 4 and the red portion is labeled 6. The width of the red rectangle is labeled 2. The first line says A sub figure equals A sub rectangle plus A sub red rectangle. Below this is A sub figure equals bh plus red bh. Below this is A sub figure equals 12 times 4 plus red 2 times 6. Below this is A sub figure equals 48 plus red 12. Below this is A sub figure equals 60.

The area of the figure is $60$ square units.

Is there another way to split this figure into two rectangles? Try it, and make sure you get the same area.

TRY IT 4.1

Find the area of each shaded region:

A blue geometric shape is shown. It looks like a horizontal rectangle attached to a vertical rectangle. The top is labeled as 8, the width of the horizontal rectangle is labeled as 2. The side is labeled as 6, the width of the vertical rectangle is labeled as 3.

Answer

28 sq. units

TRY IT 4.2

Find the area of each shaded region:

A blue geometric shape is shown. It looks like a horizontal rectangle attached to a vertical rectangle. The top is labeled as 14, the width of the horizontal rectangle is labeled as 5. The side is labeled as 10, the width of the missing space is labeled as 6.

Answer

110 sq. units

EXAMPLE 5

Find the area of the shaded region.

A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7.

Solution

We can break this irregular figure into a triangle and rectangle. The area of the figure will be the sum of the areas of triangle and rectangle.

The rectangle has a length of $8$ units and a width of $4$ units.

We need to find the base and height of the triangle.

Since both sides of the rectangle are $4$ , the vertical side of the triangle is $3$ , which is $7-4$ .

The length of the rectangle is $8$ , so the base of the triangle will be $3$ , which is $8-4$ .

A geometric shape is shown. It is a blue rectangle with a red triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7. The right side of the rectangle is labeled 4. The right side and bottom of the triangle are labeled 3.

Now we can add the areas to find the area of the irregular figure.

The top line reads A sub figure equals A sub rectangle plus A sub red triangle. The second line reads A sub figure equals lw plus one-half red bh. The next line says A sub figure equals 8 times 4 plus one-half times red 3 times red 3. The next line reads A sub figure equals 32 plus red 4.5. The last line says A sub figure equals 36.5 sq. units.

The area of the figure is $36.5$ square units.

TRY IT 5.1

Find the area of each shaded region.

A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the lower right side. The base of the rectangle is labeled 8, the height of the rectangle is labeled 4. The distance from the top of the rectangle to where the triangle begins is labeled 3, the top of the triangle is labeled 3.

Answer

36.5 sq. units

TRY IT 5.2

Find the area of each shaded region.

A blue geometric shape is shown. It looks like a rectangle with an equilateral triangle attached to the top. The base of the rectangle is labeled 12, each side is labeled 5. The base of the triangle is split into two pieces, each labeled 2.5.

Answer

70 sq. units

EXAMPLE 6

A high school track is shaped like a rectangle with a semi-circle (half a circle) on each end. The rectangle has length $105$ metres and width $68$ metres. Find the area enclosed by the track. Round your answer to the nearest hundredth.

A track is shown, shaped like a rectangle with a semi-circle attached to each side.

Solution

We will break the figure into a rectangle and two semi-circles. The area of the figure will be the sum of the areas of the rectangle and the semicircles.

A blue geometric shape is shown. It looks like a rectangle with a semi-circle attached to each side. The base of the rectangle is labeled 105 m. The height of the rectangle and diametre of the circle on the left is labeled 68 m.

The rectangle has a length of $105$ m and a width of $68$ m. The semi-circles have a diametre of $68$ m, so each has a radius of $34$ m.

TRY IT 6.1

Find the area:

A shape is shown. It is a blue rectangle with a portion of the rectangle missing. There is a red circle the same height as the rectangle attached to the missing side of the rectangle. The top of the rectangle is labeled 15, the height is labeled 9.

Answer

103.2 sq. units

TRY IT 6.2

Find the area:

A blue geometric shape is shown. It appears to be two trapezoids with a semicircle at the top. The base of the semicircle is labeled 5.2. The height of the trapezoids is labeled 6.5. The combined base of the trapezoids is labeled 3.3.

Answer

38.24 sq. units

Access Additional Online Resources

Key Concepts

Problem Solving Strategy for Geometry Applications
1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Properties of Circles

$d=2r$
Circumference: $C=2\pi r$ or $C=\pi d$
Area: $A=\pi {r}^{2}$

Glossary

irregular figure: An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas.

Practice Makes Perfect

Use the Properties of Circles

In the following exercises, solve using the properties of circles.

1. The lid of a paint bucket is a circle with radius $7$ inches. Find the a) circumference and b) area of the lid.	2. An extra-large pizza is a circle with radius $8$ inches. Find the a) circumference and b) area of the pizza.
3. A farm sprinkler spreads water in a circle with radius of $8.5$ feet. Find the a) circumference and b) area of the watered circle.	4. A circular rug has radius of $3.5$ feet. Find the a) circumference and b) area of the rug.
5. A reflecting pool is in the shape of a circle with diametre of $20$ feet. What is the circumference of the pool?	6. A turntable is a circle with diametre of $10$ inches. What is the circumference of the turntable?
7. A circular saw has a diametre of $12$ inches. What is the circumference of the saw?	8. A round coin has a diametre of $3$ centimetres. What is the circumference of the coin?
9. A barbecue grill is a circle with a diametre of $2.2$ feet. What is the circumference of the grill?	10. The top of a pie tin is a circle with a diametre of $9.5$ inches. What is the circumference of the top?
11. A circle has a circumference of $163.28$ inches. Find the diametre.	12. A circle has a circumference of $59.66$ feet. Find the diametre.
13. A circle has a circumference of $17.27$ metres. Find the diametre.	14. A circle has a circumference of $80.07$ centimetres. Find the diametre.

In the following exercises, find the radius of the circle with given circumference.

15. A circle has a circumference of $150.72$ feet.	16. A circle has a circumference of $251.2$ centimetres.
17. A circle has a circumference of $40.82$ miles.	18. A circle has a circumference of $78.5$ inches.

Find the Area of Irregular Figures

In the following exercises, find the area of the irregular figure. Round your answers to the nearest hundredth.

19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.
31.	32.
33.	34.
35.	36.
37.	38.

In the following exercises, solve.

39. A city park covers one block plus parts of four more blocks, as shown. The block is a square with sides $250$ feet long, and the triangles are isosceles right triangles. Find the area of the park.

A square is shown with four triangles coming off each side.

40. A gift box will be made from a rectangular piece of cardboard measuring $12$ inches by $20$ inches, with squares cut out of the corners of the sides, as shown. The sides of the squares are $3$ inches. Find the area of the cardboard after the corners are cut out.

A rectangle is shown. Each corner has a gray shaded square. There are dotted lines drawn across the side of each square attached to the next square.

41. Perry needs to put in a new lawn. His lot is a rectangle with a length of $120$ feet and a width of $100$ feet. The house is rectangular and measures $50$ feet by $40$ feet. His driveway is rectangular and measures $20$ feet by $30$ feet, as shown. Find the area of Perry’s lawn.

A rectangular lot is shown. In it is a home shaped like a rectangle attached to a rectangular driveway.

42. Denise is planning to put a deck in her back yard. The deck will be a $\text{20-ft}$ by $\text{12-ft}$ rectangle with a semicircle of diametre $6$ feet, as shown below. Find the area of the deck.

A picture of a deck is shown. It is shaped like a rectangle with a semi-circle attached to the top on the left side.

Everyday Math

43. Area of a Tabletop Yuki bought a drop-leaf kitchen table. The rectangular part of the table is a $\text{1-ft}$ by $\text{3-ft}$ rectangle with a semicircle at each end, as shown. a) Find the area of the table with one leaf up. b) Find the area of the table with both leaves up.

An image of a table is shown. There is a rectangular portion attached to a semi-circular portion. There is another semi-circular leaf folded down on the other side of the rectangle.

44. Painting Leora wants to paint the nursery in her house. The nursery is an $\text{8-ft}$ by $\text{10-ft}$ rectangle, and the ceiling is $8$ feet tall. There is a $\text{3-ft}$ by $\text{6.5-ft}$ door on one wall, a $\text{3-ft}$ by $\text{6.5-ft}$ closet door on another wall, and one $\text{4-ft}$ by $\text{3.5-ft}$ window on the third wall. The fourth wall has no doors or windows. If she will only paint the four walls, and not the ceiling or doors, how many square feet will she need to paint?

Writing Exercises

45. Describe two different ways to find the area of this figure, and then show your work to make sure both ways give the same area.

A geometric shape is shown. It is a vertical rectangle attached to a horizontal rectangle. The width of the vertical rectangle is 3, the left side is labeled 6, the bottom is labeled 9, and the width of the horizontal rectangle is labeled 3. The top of the horizontal rectangle is labeled 6, and the distance from the top of that rectangle to the top of the other rectangle is labeled 3.

46. A circle has a diametre of $14$ feet. Find the area of the circle a) using $3.14$ for $\pi$ b) using $\frac{22}{7}$ for $\pi$ . c) Which calculation to do prefer? Why?

Answers

1. a) 43.96 in. b) 153.86 sq. in.	3. a) 53.38 ft b) 226.865 sq. ft	5. 62.8 ft
7. 37.68 in.	9. 6.908 ft	11. 52 in.
13. 5.5 m	15. 24 ft	17. 6.5 mi
19. 16 sq. units	21. 30 sq. units	23. 57.5 sq. units
25. 12 sq. units	27. 67.5 sq. units	29. 89 sq. units
31. 44.81 sq. units	33. 41.12 sq. units	35. 35.13 sq. units
37. 95.625 sq. units	39. 187,500 sq. ft	41. 9400 sq. ft
43. a) 6.5325 sq. ft b) 10.065 sq. ft	45. Answers will vary.

Attributions

This chapter has been adapted from “Solve Geometry Applications: Circles and Irregular Figures” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

4.5 Chapter Review

Review Exercises

Systems of Measurement

In the following exercises, convert between Imperial units. Round to the nearest tenth.

1. A picture frame is $42$ inches wide. Convert the width to feet.	2. A floral arbor is $7$ feet tall. Convert the height to inches.
3. A playground is $45$ feet wide. Convert the width to yards.	4. Kelly is $5$ feet $4$ inches tall. Convert her height to inches.
5. An orca whale in the Salish Sea weighs $4.5$ tons. Convert the weight to pounds.	6. The height of Mount Shasta is $14,179$ feet. Convert the height to miles.
7. How many tablespoons are in a quart?	8. The play lasted $1\frac{3}{4}$ hours. Convert the time to minutes.
9. Trinh needs $30$ cups of paint for her class art project. Convert the volume to gallons.	10. Naomi’s baby weighed $5$ pounds $14$ ounces at birth. Convert the weight to ounces.

In the following exercises, solve, and state your answer in mixed units.

11. Every day last week, Pedro recorded the amount of time he spent reading. He read for $50,25,83,45,32,60,\text{and}\phantom{\rule{0.2em}{0ex}}135$ minutes. How much time, in hours and minutes, did Pedro spend reading?	12. John caught $4$ lobsters. The weights of the lobsters were $1$ pound $9$ ounces, $1$ pound $12$ ounces, $4$ pounds $2$ ounces, and $2$ pounds $15$ ounces. What was the total weight of the lobsters?
13. Dalila wants to make pillow covers. Each cover takes $30$ inches of fabric. How many yards and inches of fabric does she need for $4$ pillow covers?	14. Fouad is $6$ feet $2$ inches tall. If he stands on a rung of a ladder $8$ feet $10$ inches high, how high off the ground is the top of Fouad’s head?

In the following exercises, convert between metric units.

15. Mount Everest is $8,850$ metres tall. Convert the height to kilometres.	16. Donna is $1.7$ metres tall. Convert her height to centimetres.
17. One cup of yogurt contains $13$ grams of protein. Convert this to milligrams.	18. One cup of yogurt contains $488$ milligrams of calcium. Convert this to grams.
19. A bottle of water contained $650$ millilitre s. Convert this to litres.	20. Sergio weighed $2.9$ kilograms at birth. Convert this to grams.

In the following exercises, solve.

21. Selma had a $\text{1-liter}$ bottle of water. If she drank $145$ millilitres, how much water, in millilitres, was left in the bottle?	22. Minh is $2$ metres tall. His daughter is $88$ centimetres tall. How much taller, in metres, is Minh than his daughter?
23. One ounce of tofu provides $2$ grams of protein. How many milligrams of protein are provided by $5$ ounces of tofu?	24. One serving of cranberry juice contains $30$ grams of sugar. How many kilograms of sugar are in $30$ servings of cranberry juice?

In the following exercises, convert between Imperial and metric units. Round to the nearest tenth.

25. A college basketball court is $84$ feet long. Convert this length to metres.	26. Majid is $69$ inches tall. Convert his height to centimetres.
27. Lucas weighs $78$ kilograms. Convert his weight to pounds.	28. Caroline walked $2.5$ kilometres. Convert this length to miles.
29. A box of books weighs $25$ pounds. Convert this weight to kilograms.	30. Steve’s car holds $55$ litres of gas. Convert this to gallons.

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

31. $23$ °F	32. $95$ °F
33. $64$ °F	34. $20$ °F

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

35. $-5$ °C	36. $30$ °C
37. $24$ °C	38. $-12$ °C

Understand Linear, Square, Cubic Measure

In the following exercises, would you measure each item using linear, square, or cubic measure?

39. amount of sand in a sandbag	40. height of a tree
41. size of a patio	42. length of a highway

In the following exercises, find a) the perimeter b) the area of each figure

43.

Three squares are shown, in a sideways L shape.

44.

Five squares are shown, in a T-shape. There are three squares across the top and three squares down.

Use Properties of Rectangles

In the following exercises, find the a) perimeter b) area of each rectangle

45. The length of a rectangle is $42$ metres and the width is $28$ metres.	46. The length of a rectangle is $36$ feet and the width is $19$ feet.
47. A sidewalk in front of Kathy’s house is in the shape of a rectangle $4$ feet wide by $45$ feet long.	48. A rectangular room is $16$ feet wide by $12$ feet long.

In the following exercises, solve.

49. Find the length of a rectangle with perimeter of $220$ centimetres and width of $85$ centimetres.	50. Find the width of a rectangle with perimeter $39$ and length $11$ .
51. The area of a rectangle is $2356$ square metres. The length is $38$ metres. What is the width?	52. The width of a rectangle is $45$ centimetres. The area is $2700$ square centimetres. What is the length?
53. The length of a rectangle is $12$ centimetres more than the width. The perimeter is $74$ centimetres. Find the length and the width.	54. The width of a rectangle is $3$ more than twice the length. The perimeter is $96$ inches. Find the length and the width.

Use Properties of Triangles

In the following exercises, solve using the properties of triangles.

55. Find the area of a triangle with base $18$ inches and height $15$ inches.	56. Find the area of a triangle with base $33$ centimetres and height $21$ centimetres.
57. A triangular road sign has base $30$ inches and height $40$ inches. What is its area?	58. If a triangular courtyard has sides $9$ feet and $12$ feet and the perimeter is $32$ feet, how long is the third side?
59. A tile in the shape of an isosceles triangle has a base of $6$ inches. If the perimeter is $20$ inches, find the length of each of the other sides.	60. Find the length of each side of an equilateral triangle with perimeter of $81$ yards.
61. The perimeter of a triangle is $59$ feet. One side of the triangle is $3$ feet longer than the shortest side. The third side is $5$ feet longer than the shortest side. Find the length of each side.	62. One side of a triangle is three times the smallest side. The third side is $9$ feet more than the shortest side. The perimeter is $39$ feet. Find the lengths of all three sides.

Use Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

63. The height of a trapezoid is $8$ feet and the bases are $11$ and $14$ feet. What is the area?	64. The height of a trapezoid is $5$ yards and the bases are $7$ and $10$ yards. What is the area?
65. Find the area of the trapezoid with height $25$ metres and bases $32.5$ and $21.5$ metres.	66. A flag is shaped like a trapezoid with height $62$ centimetres and the bases are $91.5$ and $78.1$ centimetres. What is the area of the flag?

Use Properties of Circles

In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth.

67. A circular mosaic has radius $3$ metres. Find the

a) circumference

b) area of the mosaic

68. A circular fountain has radius $8$ feet. Find the

a) circumference

b) area of the fountain

69. Find the diametre of a circle with circumference $150.72$ inches.

70. Find the radius of a circle with circumference $345.4$ centimetres

Find the Area of Irregular Figures

In the following exercises, find the area of each shaded region.

71.	72.
73.	74.
75.	76.

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find the a) volume b) surface area of the rectangular solid

77. A rectangular solid with length $14$ centimetres, width $4.5$ centimetres, and height $10$ centimetres	78. A cube with sides that are $3$ feet long
79. A cube of tofu with sides $2.5$ inches	80. A rectangular carton with length $32$ inches, width $18$ inches, and height $10$ inches

Find Volume and Surface Area of Spheres

In the following exercises, find the a) volume b) surface area of the sphere.

81. a sphere with radius $4$ yards	82. a sphere with radius $12$ metres
83. a baseball with radius $1.45$ inches	84. a soccer ball with radius $22$ centimetres

Find Volume and Surface Area of Cylinders

In the following exercises, find the a) volume b) surface area of the cylinder

85. A cylinder with radius $2$ yards and height $6$ yards	86. A cylinder with diametre $18$ inches and height $40$ inches
87. A juice can with diametre $8$ centimetres and height $15$ centimetres	88. A cylindrical pylon with diametre $0.8$ feet and height $2.5$ feet

Find Volume of Cones

In the following exercises, find the volume of the cone.

89. A cone with height $5$ metres and radius $1$ metre	90. A cone with height $24$ feet and radius $8$ feet
91. A cone-shaped water cup with diametre $2.6$ inches and height $2.6$ inches	92. A cone-shaped pile of gravel with diametre $6$ yards and height $5$ yards

Review Answers

1. 3.5 feet	3. 15 yards	5. 9000 pounds
7. 64 tablespoons	9. 1.9 gallons	11. 7 hours 10 minutes
13. 3 yards, 12 inches	15. 8.85 kilometres	17. 13,000 milligrams
19. 0.65 litres	21. 855 millilitre s	23. 10,000 milligrams
25. 25.6 metres	27. 171.6 pounds	29. 11.4 kilograms
31. $-5$ °C	33. $17.8$ °C	35. $23$ °F
37. $75.2$ °F	39. cubic	41. square
43. a) 8 units b) 3 sq. units	45. a) 140 m b) 1176 sq. m	47. a) 98 ft. b) 180 sq. ft.
49. 25 cm	51. 62 m	53. 24.5 in., 12.5 in.
55. 135 sq. in.	57. 600 sq. in.	59. 7 in., 7 in.
61. 17 ft., 20 ft., 22 ft.	63. 100 sq. ft.	65. 675 sq. m
67. a) 18.84 m b) 28.26 sq. m	69. 48 in.	71. 30 sq. units
73. 300 sq. units	75. 199.25 sq. units	77. a) 630 cu. cm b) 496 sq. cm
79. a) 15.625 cu. in. b) 37.5 sq. in.	81. a) 267.95 cu. yd. b) 200.96 sq. yd.	83. a) 12.76 cu. in. b) 26.41 sq. in.
85. a) 75.36 cu. yd. b) 100.48 sq. yd.	87. a) 753.6 cu. cm b) 477.28 sq. cm	89. 5.233 cu. m
91. 4.599 cu. in.

Practice Test

In the following exercises, solve using the appropriate unit conversions.

1. One cup of milk contains $276$ milligrams of calcium. Convert this to grams. $\text{(1 milligram}=\text{0.001 gram)}$	2. Azize walked $4\frac{1}{2}$ miles. Convert this distance to feet. $\text{(1 mile}=\text{5,280 feet).}$
3. Janice ran $15$ kilometres. Convert this distance to miles. Round to the nearest hundredth of a mile. $\text{(1 mile}=\text{1.61 kilometres)}$	4. Larry had $5$ phone customer phone calls yesterday. The calls lasted $28,44,9,75,\text{and}\phantom{\rule{0.2em}{0ex}}55$ minutes. How much time, in hours and minutes, did Larry spend on the phone? $\text{(1 hour}=\text{60 minutes)}$
5. Use the formula $F=\frac{9}{5}C+32$ to convert $35$ °C to degrees $\text{F}$	6. Yolie is $63$ inches tall. Convert her height to centimetres. Round to the nearest centimetre. $\text{(1 inch}=\text{2.54 centimetres)}$
7. A triangular poster has base $80$ centimetres and height $55$ centimetres. Find the area of the poster.	8. The length of a rectangle is $2$ feet more than five times the width. The perimeter is $40$ feet. Find the dimensions of the rectangle.
9. A circular pool has diametre $90$ inches. What is its circumference? Round to the nearest tenth.	10. A trapezoid has height $14$ inches and bases $20$ inches and $23$ inches. Find the area of the trapezoid.
11. Find the volume of a rectangular room with width $12$ feet, length $15$ feet, and height $8$ feet.	12. Find the area of the shaded region. Round to the nearest tenth.
13. A traffic cone has height $75$ centimetres. The radius of the base is $20$ centimetres. Find the volume of the cone. Round to the nearest tenth.	14. A coffee can is shaped like a cylinder with height $7$ inches and radius $5$ inches. Find (a) the surface area and (b) the volume of the can. Round to the nearest tenth.

Practice Test Answers

1. .276 grams	2. 23760 feet	3. 9.317 miles
4. 211 minutes, 3 hours and 31 minutes	5. 95°F	6. 160 centimetres
7. 2,200 square centimetres	8. 11 feet, 9 feet	9. 282.6 inches
10. 201 feet	11. 1,440 cubic feet	12. 10.3 square inches
13. 31,400 cubic inches	14. a) 534.1 square inches b) 1335 cubic inches

III

CHAPTER 5 Trigonometry

Riverpole by Vaughn Warren – Kamloops, BC.

Trigonometry is a part of geometry that takes its origin in the ancient study of the relationship of the sides and angles of a right triangle. “Trigon” from Greek means triangle and “metron” means measure.

Applications of trigonometry are essential to many disciplines like carpentry, engineering, surveying, and astronomy, just to name a few.

How tall is the Riverpole? Do we have to climb the pole to find out? Fortunately, with the knowledge of trigonometry, we can find out the measurements of tall objects without too much hassle.

In this chapter we will explore the basic properties of angles and triangles, and the applications of the Pythagorean Theorem and trigonometric ratios.

5.1 Use Properties of Angles, Triangles, and the Pythagorean Theorem

Learning Objectives

By the end of this section, you will be able to:

Use the properties of angles
Use the properties of triangles
Use the Pythagorean Theorem

Use the Properties of Angles

Are you familiar with the phrase ‘do a $180\text{'?}$ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is $180$ degrees. See (Figure 1).

Figure 1

An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In (Figure 2), $\angle A$ is the angle with vertex at point $A$ . The measure of $\angle A$ is written $m\angle A$ .

$\angle A$ is the angle with vertex at $\text{point}\phantom{\rule{0.2em}{0ex}}A$ .

The image is an angle made up of two rays. The angle is labeled with letter A.

Figure 2

We measure angles in degrees, and use the symbol ° to represent degrees. We use the abbreviation $m$ to for the measure of an angle. So if $\angle A$ is 27°, we would write $m\angle A=27$ .

If the sum of the measures of two angles is $\text{180}$ °, then they are called supplementary angles. In (Figure 3), each pair of angles is supplementary because their measures add to $\text{180}$ °. Each angle is the supplement of the other.

The sum of the measures of supplementary angles is $\text{180}$ °.

Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.

Figure 3

If the sum of the measures of two angles is $\text{90}$ °, then the angles are complementary angles. In (Figure 4), each pair of angles is complementary, because their measures add to $\text{90}$ °. Each angle is the complement of the other.

The sum of the measures of complementary angles is $\text{90}$ °.

Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.

Figure 4

Supplementary and Complementary Angles

If the sum of the measures of two angles is $\text{180}$ °, then the angles are supplementary.

If $\angle A$ and $\angle B$ are supplementary, then $m\angle A+m\angle B=\text{180}$ °.

If the sum of the measures of two angles is $\text{90}$ °, then the angles are complementary.

If $\angle A$ and $\angle B$ are complementary, then $m\angle A+m\angle B=\text{90}$ °.

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

HOW TO: Use a Problem Solving Strategy for Geometry Applications

Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for and choose a variable to represent it.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

EXAMPLE 1

An angle measures $\text{40}$ °. Find a) its supplement, and b) its complement.

Solution

a)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

b)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 1.1

An angle measures $\text{25}$ °. Find its: a) supplement b) complement.

Answer

155°
65°

TRY IT 1.2

An angle measures $\text{77}$ °. Find its: a) supplement b) complement.

Answer

103°
13°

Did you notice that the words complementary and supplementary are in alphabetical order just like $90$ and $180$ are in numerical order?

EXAMPLE 2

Two angles are supplementary. The larger angle is $\text{30}$ ° more than the smaller angle. Find the measure of both angles.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it. The larger angle is 30° more than the smaller angle.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 2.1

Two angles are supplementary. The larger angle is $\text{100}$ ° more than the smaller angle. Find the measures of both angles.

Answer

40°, 140°

TRY IT 2.2

Two angles are complementary. The larger angle is $\text{40}$ ° more than the smaller angle. Find the measures of both angles.

Answer

25°, 65°

Use the Properties of Triangles

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in (Figure 5) is called $\Delta ABC$ , read ‘triangle $\text{ABC}$ ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

$\Delta ABC$ has vertices $A,B,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C$ and sides $a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c\text{.}$

The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.

Figure 5

The three angles of a triangle are related in a special way. The sum of their measures is $\text{180}$ °.

$m\angle A+m\angle B+m\angle C=\text{180}$ °

Sum of the Measures of the Angles of a Triangle

For any $\Delta ABC$ , the sum of the measures of the angles is $\text{180}$ °.

$m\angle A+m\angle B+m\angle C=\text{180}$ °

EXAMPLE 3

The measures of two angles of a triangle are $\text{55}$ ° and $\text{82}$ °. Find the measure of the third angle.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 3.1

The measures of two angles of a triangle are $\text{31}$ ° and $\text{128}$ °. Find the measure of the third angle.

Answer

21°

TRY IT 3.2

A triangle has angles of $\text{49}$ ° and $\text{75}$ °. Find the measure of the third angle.

Answer

56°

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one $\text{90}$ ° angle, which is often marked with the symbol shown in (Figure 6).

A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.

Figure 6

If we know that a triangle is a right triangle, we know that one angle measures $\text{90}$ ° so we only need the measure of one of the other angles in order to determine the measure of the third angle.

EXAMPLE 4

One angle of a right triangle measures $\text{28}$ °. What is the measure of the third angle?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 4.1

One angle of a right triangle measures $\text{56}$ °. What is the measure of the other angle?

Answer

34°

TRY IT 4.2

One angle of a right triangle measures $\text{45}$ °. What is the measure of the other angle?

Answer

45°

In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.

EXAMPLE 5

The measure of one angle of a right triangle is $\text{20}$ ° more than the measure of the smallest angle. Find the measures of all three angles.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the measures of all three angles
Step 3. Name. Choose a variable to represent it. Now draw the figure and label it with the given information.
Step 4. Translate. Write the appropriate formula and substitute into the formula.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 5.1

The measure of one angle of a right triangle is $\text{50}$ ° more than the measure of the smallest angle. Find the measures of all three angles.

Answer

20°, 70°, 90°

TRY IT 5.2

The measure of one angle of a right triangle is $\text{30}$ ° more than the measure of the smallest angle. Find the measures of all three angles.

Answer

30°, 60°, 90°

Similar Triangles

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.

The two triangles in (Figure 7) are similar. Each side of $\Delta ABC$ is four times the length of the corresponding side of $\Delta XYZ$ and their corresponding angles have equal measures.

$\Delta ABC$ and $\Delta XYZ$ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.

Figure 7

Properties of Similar Triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $\Delta ABC\text{:}$

$\phantom{\rule{2em}{0ex}}\begin{array}{c}\text{the length}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}BC\hfill \\ \text{the length}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}AC\hfill \\ \text{the length}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}AB\hfill \end{array}$

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

EXAMPLE 6

$\Delta ABC$ and $\Delta XYZ$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	The figure is provided.
Step 2. Identify what you are looking for.	The length of the sides of similar triangles
Step 3. Name. Choose a variable to represent it.	Let a = length of the third side of $\Delta ABC$ y = length of the third side $\Delta XYZ$
Step 4. Translate.	The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$ Since the side $AB=4$ corresponds to the side $XY=3$ , we will use the ratio $\frac{\mathrm{AB}}{\mathrm{XY}}=\frac{4}{3}$ to find the other sides. Be careful to match up corresponding sides correctly.
Step 5. Solve the equation.
Step 6. Check.
Step 7. Answer the question.	The third side of $\Delta ABC$ is 6 and the third side of $\Delta XYZ$ is 2.4.

TRY IT 6.1

$\Delta ABC$ is similar to $\Delta XYZ$ . Find $a$ .

Answer

TRY IT 6.2

$\Delta ABC$ is similar to $\Delta XYZ$ . Find $y$ .

Answer

22.5

Use the Pythagorean Theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around $500$ BCE.

Remember that a right triangle has a $\text{90}$ ° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the $\text{90}$ ° angle is called the hypotenuse, and the other two sides are called the legs. See (Figure 8).

In a right triangle, the side opposite the $\text{90}$ ° angle is called the hypotenuse and each of the other sides is called a leg.

Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”

Figure 8

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

The Pythagorean Theorem

In any right triangle $\Delta ABC$ ,

${a}^{2}+{b}^{2}={c}^{2}$

where $c$ is the length of the hypotenuse $a$ and $b$ are the lengths of the legs.

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.

To solve problems that use the Pythagorean Theorem, we will need to find square roots. We defined the notation $\sqrt{m}$ in this way:

$\text{If}\phantom{\rule{0.2em}{0ex}}m={n}^{2},\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\sqrt{m}=n\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}n\ge 0$

For example, we found that $\sqrt{25}$ is $5$ because ${5}^{2}=25$ .

We will use this definition of square roots to solve for the length of a side in a right triangle.

EXAMPLE 7

Use the Pythagorean Theorem to find the length of the hypotenuse.

Right triangle with legs labeled as 3 and 4.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length of the hypotenuse of the triangle
Step 3. Name. Choose a variable to represent it.	Let $c=\text{the length of the hypotenuse}$
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the hypotenuse is 5.

TRY IT 7.1

Use the Pythagorean Theorem to find the length of the hypotenuse.

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.

Answer

TRY IT 7.2

Use the Pythagorean Theorem to find the length of the hypotenuse.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.

Answer

EXAMPLE 8

Use the Pythagorean Theorem to find the length of the longer leg.

Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	The length of the leg of the triangle
Step 3. Name. Choose a variable to represent it.	Let $b=\text{the leg of the triangle}$ Label side b
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify.
Step 6. Check:
Step 7. Answer the question.	The length of the leg is 12.

TRY IT 8.1

Use the Pythagorean Theorem to find the length of the leg.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled “b”.

Answer

TRY IT 8.2

Use the Pythagorean Theorem to find the length of the leg.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled “b”.

Answer

EXAMPLE 9

Kelvin is building a gazebo and wants to brace each corner by placing a $\text{10-inch}$ wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let x = the distance from the corner
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.
Step 6. Check: Yes.
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1″ from the corner.

TRY IT 9.1

John puts the base of a $\text{13-ft}$ ladder $5$ feet from the wall of his house. How far up the wall does the ladder reach?

A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.

Answer

12 feet

TRY IT 9.2

Randy wants to attach a $\text{17-ft}$ string of lights to the top of the $\text{15-ft}$ mast of his sailboat. How far from the base of the mast should he attach the end of the light string?

A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.

Answer

8 feet

Key Concepts

Supplementary and Complementary Angles
- If the sum of the measures of two angles is 180°, then the angles are supplementary.
- If $\angle A$ and $\angle B$ are supplementary, then $m\angle A+m\angle B=180$ .
- If the sum of the measures of two angles is 90°, then the angles are complementary.
- If $\angle A$ and $\angle B$ are complementary, then $m\angle A+m\angle B=90$ .
Solve Geometry Applications
1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
2. Identify what you are looking for.
3. Name what you are looking for and choose a variable to represent it.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Sum of the Measures of the Angles of a Triangle
- For any $\Delta ABC$ , the sum of the measures is 180°
- $m\angle A+m\angle B=180$
Right Triangle
- A right triangle is a triangle that has one 90° angle, which is often marked with a ⦜ symbol.
Properties of Similar Triangles
- If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

Glossary

angle: An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.

complementary angles: If the sum of the measures of two angles is $\text{90}$ °, then they are called complementary angles.

hypotenuse: The side of the triangle opposite the 90° angle is called the hypotenuse.

legs of a right triangle: The sides of a right triangle adjacent to the right angle are called the legs.

right triangle: A right triangle is a triangle that has one $\text{90}$ ° angle.

similar figures: In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.

supplementary angles: If the sum of the measures of two angles is $\text{180}$ °, then they are called supplementary angles.

triangle: A triangle is a geometric figure with three sides and three angles.

vertex of an angle: When two rays meet to form an angle, the common endpoint is called the vertex of the angle.

Practice Makes Perfect

Use the Properties of Angles

In the following exercises, find a) the supplement and b) the complement of the given angle.

1. $\text{53}$ °	2. $\text{16}$ °
3. $\text{29}$ °	4. $\text{72}$ °

In the following exercises, use the properties of angles to solve.

5. Find the supplement of a $\text{135}$ ° angle.	6. Find the complement of a $\text{38}$ ° angle.
7. Find the complement of a $27.5$ ° angle.	8. Find the supplement of a $109.5$ ° angle.
9. Two angles are supplementary. The larger angle is $\text{56}$ ° more than the smaller angle. Find the measures of both angles.	10. Two angles are supplementary. The smaller angle is $\text{36}$ ° less than the larger angle. Find the measures of both angles.
11. Two angles are complementary. The smaller angle is $\text{34}$ ° less than the larger angle. Find the measures of both angles.	12. Two angles are complementary. The larger angle is $\text{52}$ ° more than the smaller angle. Find the measures of both angles.

Use the Properties of Triangles

In the following exercises, solve using properties of triangles.

13. The measures of two angles of a triangle are $\text{26}$ ° and $\text{98}$ °. Find the measure of the third angle.	14. The measures of two angles of a triangle are $\text{61}$ ° and $\text{84}$ °. Find the measure of the third angle.
15. The measures of two angles of a triangle are $\text{105}$ ° and $\text{31}$ °. Find the measure of the third angle.	16. The measures of two angles of a triangle are $\text{47}$ ° and $\text{72}$ °. Find the measure of the third angle.
17. One angle of a right triangle measures $\text{33}$ °. What is the measure of the other angle?	18. One angle of a right triangle measures $\text{51}$ °. What is the measure of the other angle?
19. One angle of a right triangle measures $22.5$ °. What is the measure of the other angle?	20. One angle of a right triangle measures $36.5$ °. What is the measure of the other angle?
21. The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.	22. The measure of the smallest angle of a right triangle is $\text{20}$ ° less than the measure of the other small angle. Find the measures of all three angles.
23. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.	24. The angles in a triangle are such that the measure of one angle is $\text{20}$ ° more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Find the Length of the Missing Side

In the following exercises, $\Delta ABC$ is similar to $\Delta XYZ$ . Find the length of the indicated side.

25. side $b$

26. side $x$

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is $270$ miles.

A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.

27. Find the distance from Los Angeles to San Francisco.

28. Find the distance from San Francisco to Las Vegas.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

29.	30.
31.	32.

Find the Length of the Missing Side

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

33.	34.
35.	36.
37.	38.
39.	40.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

41. A $\text{13-foot}$ string of lights will be attached to the top of a $\text{12-foot}$ pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?

A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.

42. Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is $12$ feet high and $16$ feet wide. How long should the banner be to fit the garage door?

A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.

43. Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of $10$ feet. What will the length of the path be?

A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.

44. Brian borrowed a $\text{20-foot}$ extension ladder to paint his house. If he sets the base of the ladder $6$ feet from the house, how far up will the top of the ladder reach?

A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.

Answers

1. a) 127° b) 37°	3. a) 151° b) 61°	5. 45°
7. 62.5°	9. 62°, 118°	11. 62°, 28°
13. 56°	15. 44°	17. 57°
19. 67.5°	21. 45°, 45°, 90°	23. 30°, 60°, 90°
25. 12	27. 351 miles	29. 15
31. 25	33. 8	35. 12
37. 10.2	39. 8	41. 5 feet
43. 14.1 feet

Attributions

This chapter has been adapted from “Use Properties of Angles, Triangles, and the Pythagorean Theorem” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

5.2 Solve Applications: Sine, Cosine and Tangent Ratios.

Learning Objectives

By the end of this section, you will be able to:

Find missing side of a right triangle using sine, cosine, or tangent ratios
Find missing angle of a right triangle using sine, cosine, or tangent ratios
Solve applications using right angle trigonometry

Now, that we know the fundamentals of algebra and geometry associated with a right triangle, we can start exploring trigonometry. Many real life problems can be represented and solved using right angle trigonometry.

Sine, Cosine, and Tangent Ratios

We know that any right triangle has three sides and a right angle. The side opposite to the right angle is called the hypotenuse. The other two angles in a right triangle are acute angles (with a measure less than 90 degrees). One of those angles we call reference angle and we use θ (theta) to represent it.

The hypotenuse is always the longest side of a right triangle. The other two sides are called opposite side and adjacent side. The names of those sides depends on which of the two acute angles is being used as a reference angle.

Figure 1.

In the right triangle each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

EXAMPLE 1

Label the sides of the triangle and find the hypotenuse, opposite, and adjacent.

Solution

We labeled the sides with a lowercase letter to match the uppercase letter of the opposite vertex.

c is hypotenuse

a is opposite

b is adjacent

TRY IT 1.1

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

Answer

y is hypotenuse

z is opposite

x is adjacent

TRY IT 1.2

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

Triangle TRS right angled at R, showing theta at T

Answer

r is hypotenuse

t is opposite

s is adjacent

Trigonometric Ratios

Trigonometric ratios are the ratios of the sides in the right triangle. For any right triangle we can define three basic trigonometric ratios: sine, cosine, and tangent.

Let us refer to Figure 1 and define the three basic trigonometric ratios as:

Three Basic Trigonometric Ratios

sine θ = $\frac{\text{the length of the opposite side}}{\text{the length of the hypotenuse side}}$
cosine θ = $\frac{\text{the length of the adjacent side}}{\text{the length of the hypotenuse side}}$
tangent θ = $\frac{\text{the length of the opposite side}}{\text{the length of the adjacent side}}$

Where θ is the measure of a reference angle measured in degrees.

Very often we use the abbreviations for sine, cosine, and tangent ratios.

sin θ = $\frac{\text{opp}}{\text{hyp}}$
cos θ = $\frac{\text{adj}}{\text{hyp}}$
tan θ = $\frac{\text{opp}}{\text{adj}}$

Some people remember the definition of the trigonometric ratios as SOH CAH TOA.

Let’s use the $\Delta DEF$ from Example 1 to find the three ratios.

EXAMPLE 2

For the given triangle find the sine, cosine and tangent ratio.

Triangle FDE right angled at D, showing theta at F

Solution

sin θ = $\frac{\text{f}}{\text{d}}$

cos θ = $\frac{\text{e}}{\text{d}}$

tan θ = $\frac{\text{f}}{\text{e}}$

TRY IT 2.1

For the given triangle find the sine cosine and tangent ratio.

Triangle XYZ right angled at Y, showing theta at Z

Answer

sin θ = $\frac{\text{z}}{\text{y}}$

cos θ = $\frac{\text{x}}{\text{y}}$

tan θ = $\frac{\text{z}}{\text{x}}$

TRY IT 2.2

For the given triangle find the sine, cosine and tangent ratio.

Answer

sin θ = $\frac{\text{t}}{\text{r}}$

cos θ = $\frac{\text{s}}{\text{r}}$

tan θ = $\frac{\text{t}}{\text{s}}$

In Example 2, our reference angles can be $\angle E$ or $\angle F$ . Using the definition of trigonometric ratios, we can write sinE= $\frac{\text{e}}{\text{d}}$ , cosE= $\frac{\text{f}}{\text{d}}$ , and tanE= $\frac{\text{e}}{\text{f}}$ .

When calculating we will usually round the ratios to four decimal places and at the end our final answer to one decimal place unless stated otherwise.

EXAMPLE 3

For the given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.

Solution

We have two possible reference angles: R an S.

Using the definitions, the trigonometric ratios for angle R are:

sin R= $\frac{4}{5}$ = 0.8
cos R= $\frac{3}{5}$ = 0.6
tan R= $\frac{4}{3}$ = 1.3333…

Using the definitions, the trigonometric ratios for angle S:

sin S = $\frac{3}{5}$ = 0.6
cos S = $\frac{4}{5}$ = 0.8
tan S = $\frac{3}{4}$ = 0.75

TRY IT 3.1

For the given triangle find the sine, cosine, and tangent ratios. If necessary round to four decimal places.

Answer

sin F = $\frac{8}{10}$ = 0.8
cos F = $\frac{6}{10}$ =0.6
tan F = $\frac{8}{6}$ = 1.3333…
sin D = $\frac{6}{10}$ = 0.6
cos D = $\frac{8}{10}$ = 0.8
tan D = $\frac{6}{8}$ = 0.75

TRY IT 3.2

For given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.

Answer

sin A = $\frac{5}{5.8}$ = 0.8621
cos A = $\frac{3}{5.8}$ =0.5172
tan A = $\frac{5}{3}$ = 1.6667
sin C = $\frac{3}{5.8}$ = 0.5172
cos C = $\frac{5}{5.8}$ = 0.8621
tan C = $\frac{3}{5}$ = 0.6

Now, let us use a scientific calculator to find the trigonometric ratios. Can you find the sin, cos, and tan buttons on your calculator? To find the trigonometric ratios make sure your calculator is in Degree Mode.

EXAMPLE 4

Using a calculator find the trigonometric ratios. If necessary, round to 4 decimal places.

a) sin 30°

b) cos 45°

c) tan 60°

Solution

Make sure your calculator is in Degree Mode.

a) Using a calculator find that sin 30° = 0.5

b) Using a calculator find that cos 45° = 0.7071 Rounded to 4 decimal places.

c) Using a calculator find that tan 60° = 1.7321 Rounded to 4 decimal places.

TRY IT 4.1

Find the trigonometric ratios. If necessary, round to 4 decimal places.

a) sin 60°

b) cos 30°

c) tan 45°

Answer

a) sin 60° = 0.8660

b) cos 30° = 0.8660

c) tan 45° = 1

TRY IT 4.2

Find the trigonometric ratios. If necessary, round to 4 decimal places.

a) sin 35°

b) cos 67°

c) tan 83°

Answer

a) sin 35° = 0.5736

b) cos 67 ° = 0.3907

c) tan 83° = 8.1443

Finding Missing Sides of a Right Triangle

In this section you will be using trigonometric ratios to solve right triangle problems. We will adapt our problem solving strategy for trigonometry applications. In addition, since those problems will involve the right triangle, it is helpful to draw it (if the drawing is not given) and label it with the given information.We will include this in the first step of the problem solving strategy for trigonometry applications.

HOW TO: Solve Trigonometry Applications

Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.
Identify what we are looking for.
Label what we are looking for by choosing a variable to represent it.
Find the required trigonometric ratio.
Solve the ratio using good algebra techniques.
Check the answer by substituting it back into the ratio in step 4 and by making sure it makes sense in the context of the problem.
Answer the question with a complete sentence.

In the next few examples, having given the measure of one acute angle and the length of one side of the right triangle, we will solve the right triangle for the missing sides.

EXAMPLE 5

Find the missing sides. Round your final answer to two decimal places

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.	A drawing is given. Angle Y is our reference angle, y is opposite side, z is adjacent side, and x=14 is the hypotenuse.
2. Identify what we are looking for.	a) the opposite side	b) adjacent side
3.Label what we are looking for by choosing a variable to represent it.	y=?	z=?
4. Find the required trigonometric ratio.	sin 35° = $\frac{y}{14}$	cos 35° = $\frac{z}{14}$
5. Solve the ratio using good algebra techniques.	14 sin 35° = y 8.03 = y	14 cos 35° = z 11.47 = z
6. Check the answer in the problem and by making sure it makes sense.	0.57 $\stackrel{?}{=}$ 8.03 $\div$ 14 0.57 = 0.57 $\checkmark$	0.82 $\stackrel{?}{=}$ 11.47 $\div$ 14 0.82 = 0.82 $\checkmark$
7. Answer the question with a complete sentence.	The opposite side is 8.03	The adjacent side is 11.47

TRY IT 5.1

Find the missing sides. Round your final answer to one decimal place.

Answer

a = 20.2

b = 16.4

TRY IT 5.2

Find the missing sides. Round your final answer to one decimal place.

Answer

d = 3.4

f = 9.4

EXAMPLE 6

Find the hypotenuse. Round your final answer to one decimal place.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Angle S is our reference angle, s is opposite side, r = 4 is the adjacent side, and p is the hypotenuse

2. Identify what we are looking for.

the hypotenuse

3.Label what we are looking for by choosing a variable to represent it.

p=?

4. Find the required trigonometric ratio.

cos 32° = $\frac{4}{p}$

5. Solve the ratio using good algebra techniques.

0.8480 = $\frac{4}{p}$

p = 4.7170

Rounding the ratios to 4 decimal places

6. Check the answer in the problem and by making sure it makes sense.

0.8480 $\stackrel{?}{=}$ $\frac{4}{4.7170}$

0.8480 = 0.8480 $\checkmark$

7. Answer the question with a complete sentence.

The hypotenuse is 4.7

Round my final answer to one decimal place.

TRY IT 6.1..

Find the hypotenuse. Round your final answer to one decimal place.

Answer

p = 22.7

TRY IT 6.2

Find the hypotenuse. Round your final answer to one decimal place.

Answer

p = 6.5

Finding Missing Angles of a Right Triangle

Sometimes we have a right triangle with only the sides given. How can we find the missing angles? To find the missing angles, we use the inverse of the trigonometric ratios. The inverse buttons sin^-1, cos^-1, and tan^-1 are on your scientific calculator.

EXAMPLE 7

Find the angles. Round your final answer to one decimal place.

a) sin A = 0.5

b) cos B = 0.9735

c) tan C = 2.89358

Solution

Use your calculator and press the 2nd FUNCTION key and then press the SIN, COS, or TAN key

a) A = sin^-10.5

$\angle A$ = 30°

b) B = cos^-10.9735

$\angle B$ = 13.2° Rounded to one decimal place

c) C = tan^-12.89358

$\angle C$ = 70.9° Rounded to one decimal place

TRY IT 7.1

Find the angles. Round your final answer to one decimal place.

a) sin X = 1

b) cos Y = 0.375

c) tan Z = 1.676767

Answer

a) $\angle X$ = 90°

b) $\angle Y$ = 68°

c) $\angle Z$ = 59.2°

TRY IT 7.2

Find the angles. Round your final answer to one decimal place.

a) sin C = 0

b) cos D = 0.95

c) tan F = 6.3333

Answer

a) $\angle C$ = 0°

b) $\angle D$ = 18.2°

c) $\angle F$ = 81°

In the example below we have a right triangle with two sides given. Our acute angles are missing. Let us see what the steps are to find the missing angles.

EXAMPLE 8

Find the missing $\angle T$ . Round your final answer to one decimal place.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Angle T is our reference angle, t = 7 is the opposite side, s is adjacent side, and r =11 is the hypotenuse

2. Identify what we are looking for.

angle T

3.Label what we are looking for by choosing a variable to represent it.

$\angle T$ =?

4. Find the required trigonometric ratio.

sin T = $\frac{7}{11}$

5. Solve the ratio using good algebra techniques.

sin T = 0.6364

T = sin^-10.6364

$\angle T$ = 39.5239°

6. Check the answer in the problem and by making sure it makes sense.

sin 39.5239° $\stackrel{?}{=}$ 0.6364

0.6364 = 0.6364 $\checkmark$

7. Answer the question with a complete sentence.

The missing angle T is 39.5°.

TRY IT 8.1

Find the missing angle X. Round your final answer to one decimal place.

Answer

20.1°

TRY IT 8.2

Find the missing angle Z. Round your final answer to one decimal place.

Answer

69.9°

EXAMPLE 9

Find the missing angle A. Round your final answer to one decimal place.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Angle A is our reference angle, a = 9 is the opposite side, c = 5 is the adjacent side, and b is the hypotenuse

2. Identify what we are looking for.

angle A

3.Label what we are looking for by choosing a variable to represent it.

$\angle A$ =?

4. Find the required trigonometric ratio.

tan A = $\frac{9}{5}$

5. Solve the ratio using good algebra techniques.

tan A = 1.8

A = tan^-1 1.8

$\angle A$ = 60.9°

6. Check the answer in the problem and by making sure it makes sense.

tan 60.9° $\stackrel{?}{=}$ 1.8

1.8 = 1.8 $\checkmark$

7. Answer the question with a complete sentence.

The missing angle A is 60.9°.

TRY IT 9.1

Find the missing angle C. Round your final answer to one decimal place.

Answer

29.1°

TRY IT 9.2

Find the missing angle E. Round your final answer to one decimal place.

Answer

36.9°

Solving a Right Triangle

From the section before we know that any triangle has three sides and three interior angles. In a right triangle, when all six parts of the triangle are known, we say that the right triangle is solved.

EXAMPLE 10

Solve the right triangle. Round your final answer to one decimal place.

Solution

Since the sum of angles in any triangle is 180°, the measure of angle B can be easy calculated.

$\angle B$ = 180° − 90° − 42°

$\angle B$ = 48°

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.	A drawing is given. Angle A is our reference angle, a = 8 is the opposite side, b is the adjacent side, and c is the hypotenuse.
2. Identify what we are looking for.	a) adjacent side	b) hypotenuse
3.Label what we are looking for by choosing a variable to represent it.	b = ?	c = ?
4. Find the required trigonometric ratio.	tan 42° = $\frac{8}{b}$	sin 42° = $\frac{8}{c}$
5. Solve the ratio using good algebra techniques.	0.9004 = $\frac{8}{b}$ 0.9004 b = 8 b = 8.8849	0.6691 = $\frac{8}{c}$ 0.6691 c = 8 c = 11.9563
6. Check the answer in the problem and by making sure it makes sense.	tan 42 ° $\stackrel{?}{=}$ $\frac{8}{8.8849}$ 0.9 = 0.9 $\checkmark$	sin 42° $\stackrel{?}{=}$ $\frac{8}{11.9563}$ 0.6691 = 0.6691 $\checkmark$
7. Answer the question with a complete sentence.	The adjacent side is 8.9. Rounded to one decimal place.	The hypotenuse is 12

We solved the right triangle

$\angle A$ = 42°

$\angle B$ = 48°

$\angle C$ = 90°

a = 8

b = 8.9

c = 12

TRY IT 10.1

Solve the right triangle. Round your final answer to one decimal place.

$\angle A$ = 21°

$\angle B$ = 69°

$\angle C$ = 90°

Answer

a = 6

b = 15.6

c = 16.7

TRY IT 10.2

Solve the right triangle. Round your final answer to one decimal place.

$\angle A$ =16°

$\angle B$ = 74°

$\angle C$ = 90°

Answer

a = 2.9

b = 10

c = 10.4

EXAMPLE 11

Solve the right triangle. Round to two decimal places.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.	A drawing is given. Let angle D be our reference angle, d = 4 is the opposite side, f is the adjacent side, and e = 9 is the hypotenuse
2. Identify what we are looking for.	a) angle D	b) adjacent
3.Label what we are looking for by choosing a variable to represent it.	$\angle D$ =?	f = ?
4. Find the required trigonometric ratio.	sin D = $\frac{4}{9}$	4² + f² = 9²
5. Solve the ratio using good algebra techniques.	sin D = 0.4444 D = sin^-10.4444 $\angle D$ = 26.3850°	16 + f² = 81 f² = 81 – 16 f² = 65 f = square root of 65 f = 8.06
6. Check the answer in the problem and by making sure it makes sense.	sin 26.3850° $\stackrel{?}{=}$ $\frac{4}{9}$ 0.4444 =0.4444 $\checkmark$	4² + 8.06² $\stackrel{?}{=}$ 9² 81 = 81 $\checkmark$
7. Answer the question with a complete sentence.	The missing angle D is 26.39°.	The adjacent side is 8.06 Rounded to two decimal places

The missing angle F = 180° – 90° – 26.39° = 63.64°

We solved the right triangle

$\angle D$ = 26.39°

$\angle E$ = 90°

$\angle F$ = 63.61°

d = 4

e = 9

f = 8.06

TRY IT 11.1

Solve the right triangle. Round to one decimal place.

$\angle D$ = 29.3°

$\angle E$ = 90°

$\angle F$ = 60.7°

Answer

d = 9

e = 18.4

f = 16

TRY IT 11.2

Solve the right triangle. Round to one decimal place.

$\angle D$ = 45.6°

$\angle E$ = 90°

$\angle F$ = 44.4°

Answer

d = 7.1

e = 10

f = 7

Solve Applications Using Trigonometric Ratios

In the previous examples we were able to find missing sides and missing angles of a right triangle. Now, let’s use the trigonometric ratios to solve real-life problems.

Many applications of trigonometric ratios involve understanding of an angle of elevation or angle of depression.

The angle of elevation is an angle between the horizontal line (ground) and the observer’s line of sight.

The angle of depression is the angle between horizontal line (that is parallel to the ground) and the observer’s line of sight.

EXAMPLE 12

James is standing 31 metres away from the base of the Harbour Centre in Vancouver. He looks up to the top of the building at a 78° angle. How tall is the Harbour Centre?

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

Angle X is our reference angle, x is opposite side, y = 31 m is the adjacent side, and z is the hypotenuse.

2. Identify what we are looking for.

The opposite side

3.Label what we are looking for by choosing a variable to represent it.

x=?

4. Find the required trigonometric ratio.

tan 78° = $\frac{x}{31}$

5. Solve the ratio using good algebra techniques.

4.7046 = $\frac{x}{31}$

x = 145.8426

6. Check the answer in the problem and by making sure it makes sense.

4.7046 $\stackrel{?}{=}$ $\frac{145.8426}{31}$

4.7046 = 4.7046 $\checkmark$

7. Answer the question with a complete sentence.

The Harbour Centre is 145.8426 metres or rounded to 146 metres.

TRY IT 12.1

Nicole is standing 75 feet away from the base of the Living Shangri-La, the tallest building in British Columbia. She looks up to the top of the building at a 83.5° angle. How tall is the Living Shangri-La?

Answer

658.3 feet.

TRY IT 12.2

Kelly is standing 23 metres away from the base of the tallest apartment building in Prince George and looks at the top of the building at a 62° angle. How tall is the building?

Answer

43.3 metres

EXAMPLE 13

Thomas is standing at the top of the building that is 45 metres high and looks at his friend that is standing on the ground, 22 metres from the base of the building. What is the angle of depression?

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

Angle Y is our reference angle, y = 45 m is the opposite side, z = 22 m is the adjacent side, and x is the hypotenuse

2. Identify what we are looking for.

angle Y

3.Label what we are looking for by choosing a variable to represent it.

$\angle Y$ =?

4. Find the required trigonometric ratio.

tan Y = $\frac{45}{22}$

5. Solve the ratio using good algebra techniques.

tan Y = 2.0455

Y = tan ^–¹2.0455

$\angle Y$ = 63.9470°

6. Check the answer in the problem and by making sure it makes sense.

tan 63.9470° $\stackrel{?}{=}$ 2.0455

2.0455 = 2.0455 $\checkmark$

7. Answer the question with a complete sentence.

The angle of depression is 63.9470° or 64° rounded to one decimal place.

TRY IT 13.1

Hemanth is standing on the top of a cliff 250 feet above the ground and looks at his friend that is standing on the ground, 40 feet from the base of the cliff. What is the angle of depression?

Answer

80.9°

TRY IT 13.2

Klaudia is standing on the ground, 25 metres from the base of the cliff and looks up at her friend on the top of a cliff 100 metres above the ground. What is the angle of elevation?

Answer

76°

Key Concepts

Three Basic Trigonometric Ratios: (Where θ is the measure of a reference angle measured in degrees.)
- sine θ = $\frac{\text{the length of the opposite side}}{\text{the length of the hypotenuse side}}$
- cosine θ = $\frac{\text{the length of the adjacent side}}{\text{the length of the hypotenuse side}}$
- tangent θ = $\frac{\text{the length of the opposite side}}{\text{the length of the adjacent side}}$
Problem-Solving Strategy for Trigonometry Applications
1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.
2. Identify what we are looking for.
3. Label what we are looking for by choosing a variable to represent it.
4. Find the required trigonometric ratio.
5. Solve the ratio using good algebra techniques.
6. Check the answer by substituting it back into the ratio solved in step 5 and by making sure it makes sense in the context of the problem.
7. Answer the question with a complete sentence.

Practice Makes Perfect

Label the sides of the triangle.

1	2.
3. If the reference angle in Question 1 is B, Find the adjacent ?	4. If the reference angle in Question 2 is Z, find the opposite ?

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

Use your calculator to find the given ratios. Round to four decimal places if necessary:

7. $\sin {47}^{\circ}$	8. $\cos {82}^{\circ}$
9. $\tan {12}^{\circ}$	10. $\sin {30}^{\circ}$

For the given triangles, find the sine, cosine and tangent of the θ.

11.	12.
13.	14.

For the given triangles, find the missing side. Round it to one decimal place.

15. Find the hypotenuse.	16. Find b.
17. Find the opposite.	18. Find the adjacent.

For the given triangles, find the missing sides. Round it to one decimal place.

19.

20.

Solve the triangles. Round to one decimal place.

21.	22.
23.	24.
25. A surveyor stands 75 metres from the bottom of a tree and looks up at the top of the tree at a 48° angle. How tall is the tree?	26. A tree makes a shadow that is 6 metres long when the angle of elevation to the sun is 52°. How tall is the tree?
27. A ladder that is 15 feet is leaning against a house and makes a 45° angle with the ground. How far is the base of the ladder from the house?	28. Matt is flying a kite and has let out 100 feet of string. The angle of elevation with the ground is 38°. How high is his kite above the ground?
29. Marta is flying a kite and has let out 28 metres of string. If the kite is 10 metres above the ground, what is the angle of elevation?	30. An airplane takes off from the ground at the angle of 25°. If the airplane traveled 200 kilometres, how high above the ground is it?

Answers

1.	3. c	5. g is opposite , f is adjacent, and e is hypotenuse
7. 0.7314	9. 0.2126	11. sin θ = $\frac{g}{e}$ , cos θ = $\frac{f}{e}$ , tan θ = $\frac{g}{f}$
13. sin θ = $\frac{s}{r}$ , cos θ = $\frac{t}{r}$ , tan θ = $\frac{s}{t}$	15. b = 19.8	17. c = 12
19. y = 19.3, z = 8.2	21. $\angle B$ = 61° $\angle C$ = 29° $\angle D$ = 90° b = 38.5 c = 21.3 d = 44	23. $\angle T$ = 36.9° $\angle R$ = 90° $\angle S$ = 53.1° t = 15 r = 25 s = 20
25. 83.3 m	27. 10.6 ft	29. 20.9°

5.3 Chapter Review

Review Exercises

Use Properties of Angles

In the following exercises, solve using properties of angles.

1. What is the supplement of a $\text{48}$ ° angle?	2. What is the complement of a $\text{61}$ ° angle?
3. Two angles are complementary. The smaller angle is $\text{24}$ ° less than the larger angle. Find the measures of both angles.	4. Two angles are supplementary. The larger angle is $\text{45}$ ° more than the smaller angle. Find the measures of both angles.

Use Properties of Triangles

In the following exercises, solve using properties of triangles.

5. The measures of two angles of a triangle are $22$ and $85$ degrees. Find the measure of the third angle.	6. One angle of a right triangle measures $41.5$ degrees. What is the measure of the other small angle?
7. One angle of a triangle is $\text{30}$ ° more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.	8. One angle of a triangle is twice the measure of the smallest angle. The third angle is $\text{60}$ ° more than the measure of the smallest angle. Find the measures of all three angles.

In the following exercises, $\Delta ABC$ is similar to $\Delta XYZ$ . Find the length of the indicated side.

Two triangles are shown. Triangle ABC is on the left. The side across from A is labeled 21, across from B is b, and across from C is 11.2. Triangle XYZ is on the right. The side across from X is labeled x, across from Y is 10, and across from Z is 8.

9. side $x$

10. side $b$

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

11.	12.
13.	14.
15.	16.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

17. Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is $8$ feet tall and Sergio has $10$ feet of wire. How far from the base of the antenna can he attach the wire?

An image of a house is shown. A 10-foot wire is going from the roof of the house to the ground. The wire hits the house at a height of 8 feet.

18. Seong is building shelving in his garage. The shelves are $36$ inches wide and $15$ inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be?

A rectangular shelf is shown, with a diagonal drawn in from the lower left corner to the upper right corner. The side is labeled 15 inches, the top is labeled 36 inches.

Find missing side of a right triangle using sine, cosine, or tangent ratios.

19. Label the triangle and find the sine cosine and tangent of θ.

20. If reference angle in above triangle is angle T, label the triangle and find the sine, cosine, and tangent of T.

Find missing angle of a right triangle using sine, cosine, or tangent ratios.

21. Find angle M

22. Find angle L.

Solve the right triangle.

23. Solve the triangle.

24.

Solve applications using right angle trigonometry.

25. A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display, as shown below. What is the angle that the string of lights makes with the ground?

A right triangle with one leg marked 12 and hypotenuse marked 13.

26. Brian borrowed a 20 foot extension ladder to use when he paints his house. If he sets the base of the ladder 6 feet from the house, as shown below, what is the angle that the ladder makes with the ground?

A house is shown with a ladder leaning against it. The ladder is marked 20’, and the distance from the house to the base of the ladder is marked 6’.

27. John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. What is the angle between the top of the ladder and the house ?

A house is shown with a ladder leaning against it. The ladder is marked 13’, and the distance from the house to the base of the ladder is marked 5’.

28. The sun is at an angle of elevation of 35°. If Bob casts a shadow that is 6 ft long, how tall is Bob?

29. A 27 foot guy wire to a pole makes an angle of 63.7° with the ground. How high from the ground is the wire attached to the pole?

30. A lighthouse is 20 metres tall. If the observer is looking at a boat that is 30 metres away from the base of the lighthouse, what is the angle of depression?

Review Answers

1. 132°	3. 33°, 57°	5. 73°
7. 30°, 60°, 90°	9. 15	11. 26
13. 8	15. 8.1	17. 6 feet
19. sin θ = $\frac{s}{r}$ , cos θ = $\frac{t}{r}$ , tan θ = $\frac{s}{t}$	21. 55.2°	23. $\angle X$ = 90° , $\angle Y$ = 57° , $\angle Z$ = 33°x = 14, y = 11.7, z= 7.6
25. 67.4°	27. 22.6°	29. 24

Practice Test

1. What is the supplement of a $\angle 57$ ° angle?	2. Two angles are complementary. The smaller angle is $\text{16}$ ° less than the larger angle. Find the measures of both angles.
3. The measures of two angles of a triangle are $29$ and $75$ degrees. Find the measure of the third angle.	4. $\Delta BCD$ is similar to $\Delta SRT$ . Find the missing sides.
5. Use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.	6. Find the hypotenuse.
7. Find angle G.	8. Solve the triangle.
9. The sun is at an angle 28°. If Adam casts a shadow that is 7 ft long, how tall is Adam?	10. The road rises 6 metres per every 100 horizontal metres. What is the angle of elevation.

Answers

1. 123°	2. 53°, 37°	3. 76°
4. b = 14, t = 7.5	5. b = 15.3	6. d = 18.4
7. $\angle G$ = 27°	8. $\angle C$ = 36.7°, $\angle B$ = 53.3°, $\angle D$ = 90°, c = 49, b = 65.7, d= 82	9. 5.5 ft
10. 3.4°

	$\text{350 L}$
Multiply by 1, writing 1 as a fraction relating litres to kilolitres.	$\text{350 L}\cdot \dfrac{\text{1 kL}}{\text{1,000 L}}$
Simplify.	$350\cancel{\text{L}}\cdot \dfrac{\text{1 kL}}{1,000\cancel{\text{L}}}$
Move the decimal 3 units to the left.	$\text{0.35 kL}$


Multiply by 1, writing 1 as a fraction relating litres to millilitre s.
Simplify.
Move the decimal 3 units to the right.

19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.
31.	32.
33.	34.
35.	36.
37.	38.

19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.
31.	32.
33.	34.
35.	36.
37.	38.

Intermediate Algebra I

Intermediate Algebra I

Contents

About BCcampus Open Education

For Students: How to Access and Use this Textbook

Tips for Using This Textbook

CHAPTER 3 Ratio, Proportion, and Percent

3.1 Ratios and Rate

Write a Ratio as a Fraction

Ratios Involving Decimals

Applications of Ratios

Ratios of Two Measurements in Different Units

Write a Rate as a Fraction

Find Unit Rates

Find Unit Price

Translate Phrases to Expressions with Fractions

Glossary

Practice Makes Perfect

Write a Ratio as a Fraction

Write a Rate as a Fraction

Find Unit Rates

Find Unit Price

Translate Phrases to Expressions with Fractions

Everyday Math

Writing Exercises

Answers

Attributions

3.2 Understand Percent

Use the Definition of Percent

Convert Percents to Fractions and Decimals

Convert Decimals and Fractions to Percents

Key Concepts

Glossary

Practice Makes Perfect

Use the Definition of Percents

Convert Percents to Fractions and Decimals

Convert Decimals and Fractions to Percents

Writing Exercises

Answers

Attributions

3.3 Solve Proportions and their Applications

Use the Definition of Proportion

Solve Proportions

Solve Applications Using Proportions

Write Percent Equations As Proportions

Translate and Solve Percent Proportions

Key Concepts

Glossary

Practice Makes Perfect

Use the Definition of Proportion

Solve Proportions

Solve Applications Using Proportions

Write Percent Equations as Proportions

Translate and Solve Percent Proportions

Everyday Math

Writing Exercises

Answers

Attributions

3.4 Solve General Applications of Percent

Translate and Solve Basic Percent Equations

Solve Applications of Percent

Find Percent Increase and Percent Decrease

Key Concepts

Glossary

Practice Makes Perfect

Translate and Solve Basic Percent Equations

Solve Applications of Percents

Find Percent Increase and Percent Decrease

Everyday Math

Writing Exercises

Answers

Attributions

3.5 Chapter Review

Review Exercises

Write a Ratio as a Fraction

Write a Rate as a Fraction

Find Unit Rates

Find Unit Price

Translate Phrases to Expressions with Fractions

Solve General Applications of Percents

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