6.3 Proportions; Health Applications

Izabela Mazur; Kim Moshenko

6. Health Option

6.3 Proportions; Health Applications

Learning Objectives

By the end of this section it is expected that 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.

EXAMPLE 1

Write the sentence as a proportion:

72 heartbeats in 1 minute is the same as 216 heartbeats in 3 minutes.

Solution

	72 is to 1 as 216 is to 3.
Write as a proportion.	$\frac{\text{72 heartbeats}} {\text{1 minute}}=\frac{\text{216 heartbeats}}{\text{3 minutes}}$

TRY IT 1

Write the sentence as a proportion:

$\phantom{\rule{0.2em}{0ex}}5$ is to $9$ as $20$ is to $36$ .

Show answer

$\phantom{\rule{0.2em}{0ex}}\frac{5}{9}=\frac{20}{36}\phantom{\rule{0.2em}{0ex}}$

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

Determine whether each equation is a proportion:

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

Show answer

no
yes

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

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

Show answer

77

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

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

Show answer

65

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

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

Show answer

−7

Solve Applications Using Proportions

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 $5$ millilitres (ml) of acetaminophen for every $25$ pounds of the child’s weight. If Zoe weighs $80$ pounds, how many millilitres 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

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

Show answer

12 ml

EXAMPLE 7

The doctor prescribes for you to take 60 milligrams of the medication that is found in a liquid cough syrup. The label on the syrup reads 100 mg /5 mL. How much cough syrup should you take?

Solution

Identify what you are asked to find.	How much cough syrup you should take?
Choose a variable to represent it.	Let $x=$ amount of cough syrup.
Write a sentence that gives the information to find it.	If there is 100 mg in 5 mL, then 60 mg is in what amount?
Translate into a proportion.	$\frac{mg}{mL}=\frac{mg}{mL}$
Substitute given values.	$\frac{100}{5}=\frac{60}{x}$
Use the cross product.	$100 x = {5}\cdot\ {60}$
Divide by 100.	x = 3
Check if the answer is reasonable.
Yes. Since 60 mg is less then 100 mg, than 3 mL is less that 5 mL.
Write a complete sentence.	You should take 3 mL of cough syrup.

TRY IT 7

The doctor prescribes 30 milligrams of the medication found in the liquid cough syrup to your daughter. The label on the syrup reads 100 mg /5 mL. How much cough syrup you should give to your daughter?

Show answer

1.5 mL

EXAMPLE 8

For adults over 18, the recommended daily allowance for protein is 0.8 grams per 1 kilogram of body weight. How many grams of protein is an adult allowed to consume per day if your weight is 68 kilograms?

Solution

Identify what you are asked to find.	How many grams of protein is an adult allowed to consume per day?
Choose a variable to represent it.	Let $p=$ number of grams of protein.
Write a sentence that gives the information to find it.	If 0.8 gr is allowed for 1 kg, then how many grams is allowed for 68 kg?
Translate into a proportion.	$\frac{protein }{weight }=\frac{protein }{weight }$
Substitute given values.	$\frac{0.8}{1}=\frac{p}{68}$
The variable is in the denominator, so find the cross products and set them equal.	$p\cdot\ {1}={0.8}\cdot\ {68}$
Simplify.	p = 54.4
Check if the answer is reasonable.
Yes, if the allowance would be 1 gram per 1 kilogram, it would be 68 grams a day.
Write a complete sentence.	An adult over 18 is allowed to consume 54.4 grams of protein.

TRY IT 8

Base on example 8, how many grams of protein is Nicole allowed if her weight is 52 kilograms?

Show answer

41.6 grams

Write Percent Equations As Proportions

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

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

Show answer

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

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

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

Show answer

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

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

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

Show answer

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

Translate and Solve Percent Proportions

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

EXAMPLE 12

A human body is made up of mostly water. As we age, total body water content also diminishes so that by the time we are in our eighties the percent of water in our bodies has decreased to around 45%. If your grandfather weights 80 kg, what number is the 45% of his weight?

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 kg is 45% of 80 kg.

TRY IT 12

Adult male typically are composed of about 60% water. If a male weights 86 kg, what number is $\text{60\%}$ of $86?$

Show answer

51.6 kg

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

According to Wikipedia, the red blood cells of an average adult human male store collectively about 2.5 grams of iron. This represents 65% of the total iron contained in the body. Find the total amount of iron contained in the body. Round your answer to the nearest tenth.

Solution

Identify the parts of the percent proportion.	65% = percent, 2.5 grams = amount, base = ?
Restate as a proportion.	65% of what number is 2.5
Set up the proportion. Let $n=$ number.	$\frac{65}{100}=\frac{2.5}{n}$
Find the cross products and set them equal.	${65}{n}={100}\cdot\ {2.5}$
Simplify.	65 n = 250
Divide both sides by 65.	$\frac{65 n}{65}=\frac{250}{65}$
Simplify.	n = 3.8
Check if the answer is reasonable.
Yes.100% is more than 65% and 3.8 is more than 2.5.
Write a complete sentence that answers the question.	The total amount of iron in the body of an average adult human male is 3.8 grams.

TRY IT 13

Translate and solve using proportions: $\text{70\%}$ of $64$ is what number?

Show answer

44.8

EXAMPLE 14

The daily recommended intake for potassium is 4700 grams. If a medium sized banana contains 425 grams of potassium, what percent of the daily recommended intake is that? Round to one decimal place.

Solution

Identify the parts of the percent proportion.	4700 = base, 425 = amount, percent = ?
Restate as a proportion.	What percent of 4700 is 425?
Set up the proportion. Let $n=$ number.	$\frac{425}{4700}=\frac{n}{100}$
Find the cross products and set them equal.	$4700 n = {425}\cdot\ {100}$
Simplify.	4700 n = 42500
Divide both sides by 4700.	$\frac{4700 n}{4700}=\frac{42500}{4700}$
Simplify.	n = 9
Check if the answer is reasonable.
Yes. 9 is less than 10 and 425 is less than 470.
Write a complete sentence that answers the question.	425 is 9% of 4700.

TRY IT 14

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

Show answer

37.5%

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 $d$ ”.

6.3 Exercise Set

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

$4$ is to $15$ as $36$ is to $135$ .
$12$ is to $5$ as $96$ is to $40$ .

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

$\frac{7}{15}=\frac{56}{120}$
$\frac{11}{6}=\frac{21}{16}$
$\frac{12}{18}=\frac{4.99}{7.56}$
$\frac{13.5}{8.5}=\frac{31.05}{19.55}$

In the following exercises, solve each proportion.

$\frac{x}{56}=\frac{7}{8}$
$\frac{49}{63}=\frac{z}{9}$
$\frac{5}{a}=\frac{65}{117}$
$\frac{98}{154}=\frac{-7}{p}$
$\frac{a}{-8}=\frac{-42}{48}$
$\frac{2.6}{3.9}=\frac{c}{3}$
$\frac{2.7}{j}=\frac{0.9}{0.2}$
$\frac{\frac{1}{2}}{1}=\frac{m}{8}$

In the following exercises, solve the proportion problem.

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?
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?
A new energy drink advertises $106$ calories for $8$ ounces. How many calories are in $12$ ounces of the drink?
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?
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?
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
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?
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?
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?

In the following exercises, translate to a proportion.

What number is $\text{35\%}$ of $250?$
$45$ is $\text{30\%}$ of what number?
What percent of $85$ is $17?$

In the following exercises, translate and solve using proportions.

What number is $\text{65\%}$ of $180?$
$\text{17\%}$ of what number is $\text{\7.65}?$
What percent of $56$ is $14?$

Answers

$\frac{4}{15}=\frac{36}{135}$
$\frac{12}{5}=\frac{96}{40}$
yes
no
no
yes
49
47
9
-11
7
2
0.6
14.4
9 ml
114, no
159 cal
$\frac{3}{4}\phantom{\rule{0.3em}{0ex}}\text{cup}$
90 mg
162, no
420 cal
300 cal
180 mg
$\frac{n}{250}=\frac{35}{100}$
$\frac{45}{n}=\frac{30}{100}$
$\frac{17}{85}=\frac{p}{100}$
117
45
25%

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.
OER.hawaii.edu
Wikipedia
Government of Canada Statistics

License

Icon for the Creative Commons Attribution 4.0 International License

Use the Definition of Proportion

Solve Proportions

Solve Applications Using Proportions

Write Percent Equations As Proportions

Translate and Solve Percent Proportions

Key Concepts

Glossary

6.3 Exercise Set

Answers

Attributions

License

Share This Book