4.1 Ratios and Rate

Izabela Mazur

CHAPTER 4 Ratio, Proportion, and Percent

4.1 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

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) $\phantom{\rule{0.2em}{0ex}}2.3\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.69$ .

Show 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) $\phantom{\rule{0.2em}{0ex}}3.4\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.68$ .

Show 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: $1\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{5}{8}$ .

Show answer

$\frac{2}{3}$

TRY IT 3.2

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

Show answer

$\frac{9}{22}$

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

$\text{36 miles per gallon}$

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.

Show answer

$18.00/hour

TRY IT 7.2

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

Show 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.

Show answer

23.5 mpg

TRY IT 8.2

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

Show answer

28 mpg

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 $\text{\$6.99}$

Show 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 $\text{\$12.72}$

Show 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

Show answer

Brand A costs $0.12 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

Show 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.

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.

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

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

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

	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.	$\frac{1}{12}$

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

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.	$\frac{\$y}{\text{18 hours}}$

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.

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

License

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