## Solving Proportions

### Learning Outcomes

• Solve a proportion equation
• Solve a proportion application

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

Solve: ${\Large\frac{x}{63}}={\Large\frac{4}{7}}$

Solution

 ${\Large\frac{x}{63}}={\Large\frac{4}{7}}$ To isolate $x$ , multiply both sides by the LCD, $63$. $\color{red}{63}({\Large\frac{x}{63}})=\color{red}{63}({\Large\frac{4}{7}})$ Simplify. $x={\Large\frac{9\cdot\color{red}{7}\cdot4}{\color{red}{7}}}$ Divide the common factors. $x=36$ Check: To check our answer, we substitute into the original proportion. ${\Large\frac{x}{63}}={\Large\frac{4}{7}}$ Substitute $x=\color{red}{36}$ ${\Large\frac{\color{red}{36}}{63}}\stackrel{?}{=}{\Large\frac{4}{7}}$ Show common factors. ${\Large\frac{4\cdot9}{7\cdot9}}\stackrel{?}{=}{\Large\frac{4}{7}}$ Simplify. ${\Large\frac{4}{7}}={\Large\frac{4}{7}}$

### try it

In the next video we show another example of how to solve a proportion equation using the LCD.

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

Solve: ${\Large\frac{144}{a}}={\Large\frac{9}{4}}$

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.

The following video shows an example of how to solve a similar problem by using the LCD.

### example

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

### 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

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?

You could also solve this proportion by setting the cross products equal.

### example

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

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?

### try it

In the following video we show another example of how to solve an application that involves proportions.