Solving Applications of Proportions

Read this text which discusses real-world examples of how proportions are used. Complete the practice problems and check your answers. These examples are word problems that use proportions.

Answers

Example 6.45

Solution

Identify what you are asked to find.	How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.	Let $a=\mathrm{ml}$ 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

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 $c= \text{number of calories}$ .
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

Solution

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.