Chase works in the stockroom of a grocery store. It takes him 20 minutes to stock 3 shelves. He has 9 more shelves to stock. His shift ends in 1 hour. Can Chase finish stocking the shelves before his shift ends?

In this concept, you will learn how to use mental math to solve proportions.

### Using Mental Math to Solve Proportions

You can find equal ratios without having to write the multiplication or division on paper. **Mental math** is the calculation that goes on inside your head and can be used to find the unknown quantity in an equation.

Here is a proportion with an unknown quantity, . Solve the proportion by finding the value of \begin{align*}x\end{align*}

First, look at the relationship between the numerators, 25 campers to 75 campers. Think, “What can I multiply 25 by to get 75. 25 times 3 equals 75.” Remember, a **proportion** is two equal ratios. You multiply both the numerator and the denominator by the same number to get an equivalent ratio.

Next, multiply the given number of tents by 3 to find the equivalent number of tents needed for 75 campers. Think, “1 times 3 equals 3.”

You need 3 tents for 75 campers.

Here is another proportion with a missing quantity. Solve the proportion using mental math.

First, look at the fraction

. 4 is half of 8. Think, “What is half of 16? Half is 16 is 8.”The value of

is 8.### Examples

#### Example 1

Earlier, you were given a problem about Chase at the grocery store.

Chase takes 20 minutes to stock 3 shelves. Can he finish 9 shelves before his shift ends in an hour? Write a proportion and use mental math to solve for

.First, write a proportion using the known values with minutes in the numerator and shelves in the denominator.

Next, look at the relationship of the denominators. Think, “What number multiplied by 3 equals 9? 3 times 3 equals 9.”

Then, multiply the known numerator by 3 to find the unknown quantity. Think, “20 times 3 equals 60. It will take 60 minutes.”

It will take Chase 60 minutes, or 1 hour, to stock 9 shelves. Chase will finish before his shift is over.

#### Example 2

Solve the proportion using mental math: \begin{align*}\frac{2}{3}=\frac{x}{33}\end{align*}

First, look at the relationship between the denominators. Think, “What can I multiply 3 by to get 33? 3 times 11 equals 33.”

Next, multiply the given numerator by 11 to find the unknown quantity. Think “2 times 11 equals 22.”

The value of

is 22.#### Example 3

Solve the proportion using mental math:

First, look at the fraction

. Think, “1 over 4 is the same as one-fourth. One-fourth of 16 is 4.”The value of

is 4.#### Example 4

Solve the proportion using mental math:

First, look at the relationship between the denominators. Think, “What can I multiply 9 by to get 18? 9 times 2 equals 18.”

Then, multiply the given numerator to find the unknown quantity. Think, “3 times 2 equals 6.”

The value of

is 6.#### Example 5

Solve the proportion using mental math: \begin{align*}\frac{5}{15}=\frac{1}{x}\end{align*}

First, look at fraction \begin{align*}\frac{5}{15}\end{align*}

Think, “5 divided by 5 is 1 and 15 divided by 5 is 3.

can be simplified and is equal to .”The value of \begin{align*}x\end{align*}

### Review

Solve the following proportions using mental math.

- \begin{align*}\frac{1}{2}=\frac{x}{8}\end{align*}
12=x8