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.
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
When pediatricians prescribe acetaminophen to children, they prescribe [latex]5[/latex] milliliters (ml) of acetaminophen for every [latex]25[/latex] pounds of the child’s weight. If Zoe weighs [latex]80[/latex] pounds, how many milliliters of acetaminophen will her doctor prescribe?
Show Solution
Solution
Identify what you are asked to find.
How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.
Let [latex]a=[/latex] ml of acetaminophen.
Write a sentence that gives the information to find it.
If [latex]5[/latex] ml is prescribed for every [latex]25[/latex] pounds, how much will be prescribed for [latex]80[/latex] pounds?
Yes. Since [latex]80[/latex] is about [latex]3[/latex] times [latex]25[/latex], the medicine should be about [latex]3[/latex] times [latex]5[/latex].
Write a complete sentence.
The pediatrician would prescribe [latex]16[/latex] ml of acetaminophen to Zoe.
You could also solve this proportion by setting the cross products equal.
try it
example
One brand of microwave popcorn has [latex]120[/latex] calories per serving. A whole bag of this popcorn has [latex]3.5[/latex] servings. How many calories are in a whole bag of this microwave popcorn?
Show Solution
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 [latex]c=[/latex] number of calories.
Write a sentence that gives the information to find it.
If there are [latex]120[/latex] calories per serving, how many calories are in a whole bag with [latex]3.5[/latex] servings?
Yes. Since [latex]3.5[/latex] is between [latex]3[/latex] and [latex]4[/latex], the total calories should be between [latex]360 (3⋅120)[/latex] and [latex]480 (4⋅120)[/latex].
Write a complete sentence.
The whole bag of microwave popcorn has [latex]420[/latex] calories.
try it
example
Josiah went to Mexico for spring break and changed $[latex]325[/latex] dollars into Mexican pesos. At that time, the exchange rate had $[latex]1[/latex] U.S. is equal to [latex]12.54[/latex] Mexican pesos. How many Mexican pesos did he get for his trip?
Show Solution
Solution
Identify what you are asked to find.
How many Mexican pesos did Josiah get?
Choose a variable to represent it.
Let [latex]p=[/latex] number of pesos.
Write a sentence that gives the information to find it.
If [latex]\text{\$1}[/latex] U.S. is equal to [latex]12.54[/latex] Mexican pesos, then [latex]\text{\$325}[/latex] is how many pesos?
The variable is in the denominator, so find the cross products and set them equal.
[latex]p\cdot{1}=12.54(325)[/latex]
Simplify.
[latex]c=4,075.5[/latex]
Check if the answer is reasonable.
Yes, [latex]\text{\$100}[/latex] would be [latex]\text{\$1,254}[/latex] pesos. [latex]\text{\$325}[/latex] is a little more than [latex]3[/latex] times this amount.
Write a complete sentence.
Josiah has [latex]4075.5[/latex] pesos for his spring break trip.
try it
In the following video we show another example of how to solve an application that involves proportions.
This is the end of the section. Close this tab and proceed to the corresponding assignment.