Solving Applications of Proportions

Site:	Saylor Academy
Course:	RWM101: Foundations of Real World Math
Book:	Solving Applications of Proportions

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Date:	Thursday, 3 April 2025, 10:16 PM

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Solve Applications Using Proportions
Exmaples
- Answers

Let's consider more real-world examples that use proportions. Read this text, paying close attention to how percents can be written as proportions. Complete the practice problems and check your answers.

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.

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

$\begin{aligned}\dfrac{\text { amount }}{\text { base }} &=\dfrac{\text { percent }}{100} \\\dfrac{3}{5} &=\dfrac{60}{100}\end{aligned}$

PERCENT PROPORTION

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

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

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

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

We could also say:

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

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

Translate and Solve Percent Proportions

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

When the percent is more than 100, which is more than one whole, the unknown number will be more than the base.

Percents with decimals and money are also used in proportions.

Source: Rice University, https://openstax.org/books/prealgebra/pages/6-5-solve-proportions-and-their-applications
This work is licensed under a Creative Commons Attribution 4.0 License.

EXAMPLE 6.45

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?

EXAMPLE 6.46

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?

EXAMPLE 6.47

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?

Example 6.45

Identify what you are asked to find.	How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.	Let $a=\mathrm{ml}$ 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.	$\dfrac{\mathrm{ml}}{\text { pounds }}=\dfrac{\mathrm{ml}}{\text { pounds }}$
Substitute given values - be careful of the units.	$\dfrac{5}{25}=\dfrac{a}{80}$
Multiply both sides by $80$ .	$80 \cdot \dfrac{5}{25}=80 \cdot \dfrac{a}{80}$
Multiply and show common factors.	$\dfrac{16 \cdot 5 \cdot 5}{5 \cdot 5}=\dfrac{80 a}{80}$
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 6.46

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

Example 6.47

Identify what you are asked to find.	How many Mexican pesos did Josiah get?
Choose a variable to represent it.	Let $p=\text{number of pesos}$ .
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.	$\dfrac{\$}{\text { pesos }}=\dfrac{\$}{\text { pesos }}$
Substitute given values.	$\dfrac{1}{12.54}=\dfrac{325}{p}$
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 $4075.5$ pesos for his spring break trip.