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# Proportions Using Cross Products

## Cross-multiply to solve proportions with one variable

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Proportions Using Cross Products
Credit: CK-12 Foundation
Source: CK-12 Foundation
License: CC BY-NC 3.0

Jamaya dropped bottles and wrappers whenever and wherever she wanted to. In order to break this bad habit, Jamaya’s mother made her help with a local clean-up effort. For every piece of litter that Jamaya had dropped in and around her home, she had to pick up 10 pieces with the group. If Jamaya dropped a total of 15 bottles, cans, and wrappers, how many pieces of litter does she have to clean up?

In this concept, you will learn how to solve proportions using cross products.

### Solving Proportions Using Cross Products

ratio represents a comparison between two quantities. Equivalent ratios are ratios that are equal. A proportion is made up of two equivalent ratios.

Proportional reasoning, or examining the relationship between two numbers, can be used to determine the value of x.

A proportion can be expressed as two equivalent fractions.

ab=cd\begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}

A proportion can be expressed with colons.

a:b=c:d\begin{align*}a:b = c:d\end{align*}

In a proportion, the means are the two terms that are closest together when the proportion is written with colons. So, in a:b=c:d\begin{align*}a:b = c:d\end{align*}, the means are b\begin{align*}b\end{align*} and c\begin{align*}c\end{align*}.

The extremes are the terms in the proportion that are furthest apart when the proportion is written with colons. So, in a:b=c:d\begin{align*}a:b = c:d\end{align*}, the extremes are a\begin{align*}a\end{align*}and d\begin{align*}d\end{align*}.

The diagram below shows how to identify the means and the extremes in a proportion.

Credit: CK-12 Foundation
Source: CK-12 Foundation
License: CC BY-NC 3.0

The Cross Products Property of Proportions states that the product of the means is equal to the product of the extremes.

abbc=cd=ad\begin{align*}\frac{a}{b}&=\frac{c}{d}\\ b\cdot c&=a\cdot d\end{align*}

Here is an example.

Solve for a.

a4=68\begin{align*}\frac{a}{4}=\frac{6}{8}\end{align*}

First, multiply the means and the extremes and set them equal to one another.

a88a=46=24\begin{align*}a\cdot 8&=4\cdot6\\ 8a&=24\end{align*}

Next, solve the equation for the missing variable.

8a8a=248=3\begin{align*}\frac{8a}{8}&=\frac{24}{8}\\ a&=3\end{align*}

The answer is a = 3.

### Examples

#### Example 1

Earlier, you were given a problem about Jamaya and her littering habit.

She had dropped a total of 15 pieces of litter in and around her home, and her mother said she had to pick up 10 pieces for every piece that she dropped.   How many pieces of litter does Jamaya have to clean up?

First, write a proportion.

10 pieces to pick up1 piece dropped=x pieces to pick up15 pieces dropped\begin{align*}\frac{10 \ pieces \ to \ pick \ up }{1 \ piece \ dropped}= \frac{x \ pieces \ to \ pick \ up}{15 \ pieces \ dropped}\end{align*}

Next, cross multiply.

x=150\begin{align*}x = 150\end{align*}

The answer is x=150.

Jamaya must pick up 150 pieces of litter.

#### Example 2

Solve for x.

x5=610\begin{align*}\frac{x}{5}=\frac{6}{10}\end{align*}

First, cross multiply and set the products equal to one another.

10x=30\begin{align*}10x = 30\end{align*}

Next, solve for x.

x=3\begin{align*}x = 3\end{align*}

The answer is x=3.

#### Example 3

Solve for a.

a9=1527\begin{align*}\frac{a}{9}=\frac{15}{27}\end{align*}

First, cross multiply and set the products equal to one another.

27a=135\begin{align*}27a = 135\end{align*}

Next, solve for a.

a=5\begin{align*}a = 5\end{align*}

The answer is a = 5.

Example 4

Solve for b.

b4=1216\begin{align*}\frac{b}{4}=\frac{12}{16}\end{align*}

First, cross multiply and set the products equal to one another.

16b=48\begin{align*}16b = 48\end{align*}

Next, solve for b.

b=3\begin{align*}b = 3\end{align*}

The answer is b = 3.

Example 5

Rudy was a silly kitten who loved to play with ping pong balls. It took him just 15 minutes to swat 12 of them underneath the sofa. How many ping pong balls could Rudy hit under the sofa in 1 hour?

First, write a proportion.

15 minutes12 balls=60 minutesx balls\begin{align*}\frac{15 \ minutes}{12 \ balls} = \frac{60 \ minutes}{x \ balls}\end{align*}

Next, cross multiply.

15x=720\begin{align*}15x = 720\end{align*}

Then, solve for x.

x=48\begin{align*}x = 48\end{align*}

The answer is 48. In one hour, 60 minutes, Rudy could swat 48 ping pong balls under the sofa.

### Review

Use cross products to find the value of the variable in each proportion.

1. 610=x5\begin{align*}\frac{6}{10} = \frac{x} {5}\end{align*}
2. 23=x9\begin{align*}\frac{2}{3} = \frac{x} {9}\end{align*}
3. 49=a45\begin{align*}\frac{4}{9} = \frac{a} {45}\end{align*}
4. 78=a4\begin{align*}\frac{7}{8} = \frac{a} {4}\end{align*}
5. b8=516\begin{align*}\frac{b}{8} = \frac{5} {16}\end{align*}
6. 63=x9\begin{align*}\frac{6}{3} = \frac{x} {9}\end{align*}
7. 4x=810\begin{align*}\frac{4}{x} = \frac{8} {10}\end{align*}
8. 1.5y=39\begin{align*}\frac{1.5}{y} = \frac{3} {9}\end{align*}
9. 411=c33\begin{align*}\frac{4}{11} = \frac{c} {33}\end{align*}
10. 26=5y\begin{align*}\frac{2}{6} = \frac{5} {y}\end{align*}
11. 210=5x\begin{align*}\frac{2}{10} = \frac{5} {x}\end{align*}
12. 412=6n\begin{align*}\frac{4}{12} = \frac{6} {n}\end{align*}
13. 5r=70126\begin{align*}\frac{5}{r} = \frac{70} {126}\end{align*}
14. 414=14k\begin{align*}\frac{4}{14} = \frac{14} {k}\end{align*}
15. 8w=63\begin{align*}\frac{8}{w} = \frac{6} {3}\end{align*}
16. 25=17a\begin{align*}\frac{2}{5} = \frac{17} {a}\end{align*}

To see the Review answers, open this PDF file and look for section 5.10.

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### Vocabulary Language: English

Cross Product Property of Proportions

The cross product property of proportions states that the cross products of two ratios will be equal if the two ratios form a proportion.

Cross Products

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

Extremes

In a proportion, the extremes are the values of the proportion that are furthest apart when written in ratio form using a colon. For example: In the proportion a : b = c : d, a and d are the extremes.

Means

In a proportion, the means are the values of the proportion that are close to each other when written in ratio form using a colon.