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Mental Math to Solve Proportions

Solve proportions by comparing numerators and denominators of equal fractions.

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Mental Math to Solve Proportions
Credit: Yuya Tamai
Source: https://www.flickr.com/photos/tamaiyuya/5694615823/
License: CC BY-NC 3.0

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, \begin{align*}x\end{align*}x. Solve the proportion by finding the value of \begin{align*}x\end{align*}x using mental math.

\begin{align*}\frac{25 \text{ campers}}{1 \text{ tent}}=\frac{75 \text{ campers}}{x \text{ tents}}\end{align*}25 campers1 tent=75 campersx tents

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.

\begin{align*}\frac{4}{8}=\frac{x}{16}\end{align*}48=x16

First, look at the fraction \begin{align*}\frac{4}{8}\end{align*}48. 4 is half of 8. Think, “What is half of 16? Half is 16 is 8.”

The value of \begin{align*}x\end{align*}x 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 \begin{align*}x\end{align*}x.

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

\begin{align*}\frac{20}{3}=\frac{x}{9}\end{align*}203=x9

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*}23=x33

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 \begin{align*}x\end{align*}x is 22.

Example 3

Solve the proportion using mental math: \begin{align*}\frac{1}{4}=\frac{x}{16}\end{align*}14=x16.

First, look at the fraction \begin{align*}\frac{1}{4}\end{align*}14. Think, “1 over 4 is the same as one-fourth. One-fourth of 16 is 4.”

The value of \begin{align*}x\end{align*}x is 4.

Example 4

Solve the proportion using mental math: \begin{align*}\frac{3}{9}=\frac{x}{18}\end{align*}39=x18.

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 \begin{align*}x\end{align*}x is 6.

Example 5

Solve the proportion using mental math: \begin{align*}\frac{5}{15}=\frac{1}{x}\end{align*}515=1x.

First, look at fraction \begin{align*}\frac{5}{15}\end{align*}515. 5 and 15 are multiples of 5 and can be simplified by dividing both the numerator and denominator by 5.

Think, “5 divided by 5 is 1 and 15 divided by 5 is 3. \begin{align*}\frac{5}{15}\end{align*}515 can be simplified and is equal to \begin{align*}\frac{1}{3}\end{align*}13.”

The value of \begin{align*}x\end{align*}x is 3.

Review

Solve the following proportions using mental math. 

  1. \begin{align*}\frac{1}{2}=\frac{x}{8}\end{align*}12=x8
  2. \begin{align*}\frac{1}{2}=\frac{5}{x}\end{align*}12=5x
  3. \begin{align*}\frac{1}{3}=\frac{4}{x}\end{align*}13=4x
  4. \begin{align*}\frac{2}{3}=\frac{x}{6}\end{align*}23=x6
  5. \begin{align*}\frac{1}{2}=\frac{x}{16}\end{align*}12=x16
  6. \begin{align*}\frac{5}{6}=\frac{x}{12}\end{align*}56=x12
  7. \begin{align*}\frac{14}{16}=\frac{x}{8}\end{align*}1416=x8
  8. \begin{align*}\frac{1}{2}=\frac{x}{18}\end{align*}12=x18
  9. \begin{align*}\frac{1}{4}=\frac{x}{20}\end{align*}14=x20
  10. \begin{align*}\frac{1}{4}=\frac{x}{24}\end{align*}14=x24
  11. \begin{align*}\frac{1}{4}=\frac{x}{40}\end{align*}14=x40
  12. \begin{align*}\frac{2}{4}=\frac{x}{40}\end{align*}24=x40
  13. \begin{align*}\frac{25}{50}=\frac{2}{x}\end{align*}2550=2x
  14. \begin{align*}\frac{4}{12}=\frac{x}{48}\end{align*}412=x48
  15. \begin{align*}\frac{6}{7}=\frac{36}{x}\end{align*}67=36x

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.7. 

Vocabulary

Cross Products

To simplify a proportion using cross products, multiply the diagonals of each ratio.

Proportion

A proportion is an equation that shows two equivalent ratios.

Image Attributions

  1. [1]^ Credit: Yuya Tamai; Source: https://www.flickr.com/photos/tamaiyuya/5694615823/; License: CC BY-NC 3.0

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