Solve Proportions and their Applications

6.5 Solve Proportions and their Applications

The topics covered in this section are:

  1. Use the definition of proportion
  2. Solve proportions
  3. Solve applications using proportions
  4. Write percent equations as proportions
  5. Translate and solve percent proportions

6.5.1 Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. 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 $b \neq 0, d \neq 0$.

The proportion states two ratios or rates are equal. The proportion is read “$a$ is to $b$, as $c$ is to $d$”.

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 “$1$ is to $2$ as $4$ is to $8$”.

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion $\frac{20\ \mathrm{students}}{1\ \mathrm{teacher}} = \frac{60\ \mathrm{students}}{3\ \mathrm{teachers}}$ 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:

  1. $3$ is to $7$ as $15$ is to $35$.
  2. $5$ hits in $8$ at bats is the same as $30$ hits in $48$ at-bats.
  3. $ \$1.50$ for $6$ ounces is equivalent to $ \$2.25$ for $9$ ounces.
Solution

Part 1.
$3$ is to $7$ as $15$ is to $35$.
Write as a proportion.$\frac{3}{7} = \frac{15}{
35}$
Part 2.
$5$ hits in $8$ at bats is the same as $30$ hits in $48$ at-bats.
Write each fraction to compare hits to at-bats.$\frac{\mathrm{hits}}{\mathrm{at-bats}} = \frac{\mathrm{hits}}{\mathrm{at-bats}}$
Write as a proportion.$\frac{5}{8} = \frac{30}{
48}$
Part 3.
$ \$1.50$ for $6$ ounces is equivalent to $ \$2.25$ for $9$ ounces.
Write each fraction to compare dollars to ounces.$\frac{$}{\mathrm{ounces}} = \frac{$}{\mathrm{ounces}}$
Write as a proportion.$\frac{1.50}{6} = \frac{2.25}{
9}$

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.

The figure shows cross multiplication of two proportions. There is the proportion 1 is to 2 as 4 is to 8. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 8 · 1 = 8 and 2 · 4 = 8. There is the proportion 2 is to 3 as 6 is to 9. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 9 · 2 = 18 and 3 · 6 = 18.

CROSS PRODUCTS OF A PROPORTION

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

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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:

  1. $\frac{4}{9} = \frac{12}{28}$
  2. $\frac{17.5}{37.5} = \frac{7}{15}$
Solution

Part 1.
$\frac{4}{9} = \frac{12}{28}$
Find the cross products.$28 \cdot 4 = 112$ and $9 \cdot 12 =108$
.

Since the cross products are not equal, $28 \cdot 4 \neq 9 \cdot 12$, the equation is not a proportion.

Part 2.
$\frac{17.5}{37.5} = \frac{7}{15}$
Find the cross products.$15 \cdot 17.5 = 262.5$ and $37.5 \cdot 7 =262.5$
.

Since the cross products are equal, $15 \cdot 17.5 = 37.5 \cdot 7$, the equation is a proportion.

6.5.2 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

$\frac{x}{63} = \frac{4}{7}$
To isolate $x$, multiply both sides by the LCD, $63$.$63( \frac{x}{63} ) = 63( \frac{4}{7} )$
Simplify.$x= \frac{9 \cdot \cancel{7} \cdot 4}{\cancel{7}}$
Divide the common factors.$x=36$
Check: To check our answer, we substitute into the original proportion.
$\frac{x}{63} = \frac{4}{7}$
Substitute $x=36$$\frac{x}{63} \stackrel{?}{=} \frac{4}{7}$
Show common factors.$\frac{4 \cdot 9}{7 \cdot 9} \stackrel{?}{=} \frac{4}{7}$
Simplify.$\frac{4}{7} = \frac{4}{7}$

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: $\frac{114}{a} = \frac{9}{4}$.

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.$4 \cdot 144 = a \cdot 9$
Simplify.$576=9a$
Divide both sides by $9$.$\frac{576}{9} = \frac{9a}{9}$
Simplify.$64=a$
Check your answer.
$\frac{144}{a} = \frac{9}{4}$
Substitute $a=64$$\frac{144}{64} \stackrel{?}{=} \frac{9}{4}$
Show common factors.$\frac{9 \cdot 16}{4 \cdot 16} \stackrel{?}{=} \frac{9}{4}$
Simplify.$\frac{9}{4} = \frac{9}{4}$

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.

Example 5

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

Solution
Find the cross products and set them equal..
$y \cdot 52 = 91(-4)$
Simplify.$52y=-364$
Divide both sides by $52$.$\frac{52y}{52} = \frac{-364}{52}$
Simplify.$y=-7$
Check.
$\frac{52}{91} = \frac{-4}{y}$
Substitute $a=-7$$\frac{52}{91} \stackrel{?}{=} \frac{-4}{-7}$
Show common factors.$\frac{4 \cdot 13}{7 \cdot 13} \stackrel{?}{=} \frac{-4}{-7}$
Simplify.$\frac{4}{7} = \frac{4}{7}$

6.5.3 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 $5$ milliliters (ml) of acetaminophen for every $25$ pounds of the child’s weight. If Zoe weighs $80$ pounds, how many milliliters 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.$\frac{\mathrm{ml}}{\mathrm{pounds}} = \frac{\mathrm{ml}}{\mathrm{pounds}}$
Substitute given values—be careful of the units.$\frac{5}{25} = \frac{a}{80}$
Multiply both sides by $80$.$80 \cdot \frac{5}{25} = 80 \cdot \frac{a}{80}$
Multiply and show common factors.$\frac{16 \cdot 5 \cdot 5}{5 \cdot 5} = \frac{80a}{80}$
Simplify.$16=a$
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.

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.$\frac{\mathrm{calories}}{\mathrm{serving}} = \frac{\mathrm{calories}}{\mathrm{serving}}$
Substitute given values.$\frac{120}{1} = \frac{c}{3.5}$
Multiply both sides by $3.5$.$(3.5)( \frac{120}{1} ) = (3.5) ( \frac{c}{3.5} )$
Multiply.$420=c$
Check if the answer is reasonable.
Yes. Since $3.5$ is between $3$ and $4$, the total calories should be between $360 (3 \cdot 120)$ and $480 (4 \cdot 120 )$.
Write a complete sentence.The whole bag of microwave popcorn has $420$ calories.

Example 8

Josiah went to Mexico for spring break and changed $ \$ 325$ dollars into Mexican pesos. At that time, the exchange rate had $ \$ 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.$\frac{$}{\mathrm{pesos}} = \frac{$}{\mathrm{pesos}}$
Substitute given values.$\frac{1}{12.54} = \frac{325}{p}$
The variable is in the denominator, so find the cross products and set them equal.$p \cdot 1 = 12.54(325)$
Simplify.$c=4,075.5$
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 $4,075.5$ pesos for his spring break trip.

6.5.4 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, $60 \% = \frac{60}{100}$ 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:

$\large \frac{\mathrm{amount}}{\mathrm{base}} = \frac{\mathrm{percent}}{100}$

$\large \frac{3}{5} = \frac{60}{100}$

PERCENT PROPORTION

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

$\large \frac{\mathrm{amount}}{\mathrm{base}} = \frac{\mathrm{percent}}{100}$

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

𝑇ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑖𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑖𝑠 𝑡𝑜 𝑜𝑛𝑒 ℎ𝑢𝑛𝑑𝑟𝑒𝑑.

We could also say:

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 $75 \%$ 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.What number out of $90$ is the same as $75$ out of $100$?
Set up the proportion. Let $n=$ number.$\frac{n}{90} = \frac{75}{100}$

Example 10

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

Solution
Identify the parts of the percent proportion..
Restate as a proportion.$19$ out of what number is the same as $25$ out of $100$?
Set up the proportion. Let $n=$ number.$\frac{19}{n} = \frac{25}{100}$

Example 11

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

Solution
Identify the parts of the percent proportion..
Restate as a proportion.$9$ out of $27$ is the same as what number out of $100$?
Set up the proportion. Let $p=$ percent.$\frac{9}{27} = \frac{p}{100}$

6.5.5 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 $45 \%$ of $80$?

Solution

Identify the parts of the percent proportion..
Restate as a proportion.What number out of $80$ is the same as $45$ out of $100$?
Set up the proportion. Let $n=$ number.$\frac{n}{80} = \frac{45}{100}$
Find the cross products and set them equal.$100 \cdot n = 80 \cdot 45$
Simplify.$100n=3,600$
Divide both sides by $100$.$\frac{100n}{100} = \frac{3,600}{100}$
Simplify.$n=36$
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 $45 \%$ 0f $80$.

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: $125 \%$ of $25$ is what number?

Solution

Identify the parts of the percent proportion..
Restate as a proportion.What number out of $25$ is the same as $125$ out of $100$?
Set up the proportion. Let $n=$ number.$\frac{n}{25} = \frac{125}{100}$
Find the cross products and set them equal.$100 \cdot n = 25 \cdot 125$
Simplify.$100n=3,125$
Divide both sides by $100$.$\frac{100n}{100} = \frac{3,125}{100}$
Simplify.$n=31.25$
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 $31.25$.

Percents with decimals and money are also used in proportions.

Example 14

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

Solution

Identify the parts of the percent proportion..
Restate as a proportion.$ \$1.56$ out of what number is the same as $6.5$ out of $100$?
Set up the proportion. Let $n=$ number.$\frac{1.56}{n} = \frac{6.5}{100}$
Find the cross products and set them equal.$100(1.56) = n \cdot 6.5$
Simplify.$156=6.5n$
Divide both sides by $6.5$ to isolate the variable.$\frac{156}{6.5} = \frac{6.5n}{16.5}$
Simplify.$24=n$
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$.

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.$9$ out of $72$ is the same as what number out of $100$?
Set up the proportion. Let $n=$ number.$\frac{9}{72} = \frac{n}{100}$
Find the cross products and set them equal.$72 \cdot n = 100 \cdot 9$
Simplify.$72n=900$
Divide both sides by $72$.$\frac{72n}{72} = \frac{900}{72}$
Simplify.$n=12.5$
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 $9$.
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