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# Proportions Using Cross Products

## Cross-multiply to solve proportions with one variable

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Proportions Using Cross Products

Tony is training for a marathon. A marathon is about 26 miles. Last week, it took him 120 minutes to run 13 miles. If Tony runs at the same speed, how long will it take him to run a full marathon?

In this concept, you will learn to use the cross product to identify proportions.

### Using Cross Products to Identify Proportions

A proportion is two equal ratios. Proportions can be written as equations. A ratio compares two quantities. One way to check if two ratios are equivalent is to compare the simplest form of each ratio.

Compare the ratios 1:4 and 5:20. Write the ratios as fractions and simplify to compare.

14and520\begin{align*}\frac{1}{4} \quad \text{and} \quad \frac{5}{20}\end{align*}

14\begin{align*}\frac{1}{4}\end{align*} is in simplest form. 520\begin{align*}\frac{5}{20}\end{align*} can be simplified by dividing the numerator and denominator by the greatest common factor, 5.

14and5÷520÷5=14\begin{align*}\frac{1}{4} \quad \text{and} \quad \frac{5 \div 5}{20 \div 5} = \frac{1}{4}\end{align*}

The ratios 1:4 and 5:20 are equivalent ratios.

Another way to check if two ratios are equivalent is to cross multiply and compare the cross products. A cross product is the result of multiplying the numerator of one ratio with the denominator of another. If the cross products are equal, then the ratios are proportional.

For ratios ab\begin{align*}\frac{a}{b}\end{align*} and cd\begin{align*}\frac{c}{d}\end{align*}, if ad=bc\begin{align*}ad=bc\end{align*}, then ab\begin{align*}\frac{a}{b}\end{align*} and cd\begin{align*}\frac{c}{d}\end{align*} are proportional and can be written as ab=cd\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}.

Let’s use cross products to compare the ratios 23\begin{align*}\frac{2}{3}\end{align*} and 46\begin{align*}\frac{4}{6}\end{align*} to determine if they are proportional.

First, write an equation with the ratios.

23=46\begin{align*}\frac{2}{3} = \frac{4}{6}\end{align*}

Next, cross multiply to find the cross products.

\begin{align*}2 \times 6 = 3 \times 4\end{align*}

Then, simplify both sides of the equation by multiplying and check if they are equal.

\begin{align*}12=12\end{align*}

Both sides are equal. The ratios \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{4}{6}\end{align*} are proportional.

### Examples

#### Example 1

Earlier, you were given a problem about Tony training for the marathon.

Tony took 120 minutes to run 13 miles. Use cross products to find how long it will take Tony to run 26 miles if he continues at the same speed. (Note: Write the corresponding units in the same location. The minutes are in the numerator and miles are in the denominator.)

\begin{align*}\frac{\text{minutes}}{\text{miles}}=\frac{\text{minutes}}{\text{miles}}\end{align*}

First, write an equation with the ratios. Use \begin{align*}x\end{align*} to represent the unknown quantity.

\begin{align*}\frac{120}{13}=\frac{x}{26}\end{align*}

Next, cross multiply to find the cross products.

\begin{align*}120 \times 26 = 13x\end{align*}

Then, simplify by multiplying 120 and 26.

\begin{align*}3120=13x\end{align*}

Divide both sides by 13 to find the unknown quantity.

\begin{align*}\begin{array}{rcl} \frac{3120}{13} & = & \frac{13x}{13}\\ 240 & = & x \end{array}\end{align*}

It will take Tony about 240 minutes or 4 hours to run the marathon.

#### Example 2

Use the cross products to determine if the ratios \begin{align*}\frac{6}{9}\end{align*} and \begin{align*}\frac{3}{4.5}\end{align*} are proportional.

First, write an equation with the ratios.

\begin{align*}\frac{6}{9} = \frac{3}{4.5}\end{align*}

Next, cross multiply to find the cross products.

\begin{align*}6 \times 4.5 = 9 \times 3\end{align*}

Then, simplify both sides of the equation by multiplying and check if they are equal.

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

Both sides are equal. The ratios \begin{align*}\frac{6}{9}\end{align*} and \begin{align*}\frac{3}{4.5}\end{align*} are proportional.

#### Example 3

Use the cross products to determine if the ratios \begin{align*}\frac{2}{5} \quad \text{and} \quad \frac{5}{9}\end{align*} are proportional.

First, write an equation with the ratios.

\begin{align*}\frac{2}{5} = \frac{5}{9}\end{align*}

Next, cross multiply to find the cross products.

\begin{align*}2 \times 9 = 5 \times 5\end{align*}

Then, simplify both sides of the equation by multiplying and check if they are equal.

\begin{align*}18 \neq 25\end{align*}

18 is not equal to 25. \begin{align*}\frac{2}{5}\end{align*} and \begin{align*}\frac{5}{9}\end{align*} are not proportional.

#### Example 4

Use the cross products to determine if the ratios \begin{align*}\frac{3}{6} \quad \text{and} \quad \frac{5}{10}\end{align*} are proportional.

First, write an equation with the ratios.

\begin{align*}\frac{3}{6} = \frac{5}{10}\end{align*}

Next, cross multiply to find the cross products.

\begin{align*}3 \times 10 = 6 \times 5\end{align*}

Then, simplify both sides of the equation by multiplying and check if they are equal.

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

Both sides are equal. \begin{align*}\frac{3}{6}\end{align*} and \begin{align*}\frac{5}{10}\end{align*} are proportional.

#### Example 5

Use the cross products to determine if the ratios \begin{align*}\frac{4}{7} \quad \text{and} \quad \frac{12}{28}\end{align*} are proportional.

First, write an equation with the ratios.

\begin{align*}\frac{4}{7} = \frac{12}{28}\end{align*}

Next, cross multiply to find the cross products.

\begin{align*}4 \times 28 = 7 \times 12\end{align*}

Then, simplify both sides of the equation by multiplying and check if they are equal.

\begin{align*}112 \neq 84\end{align*}

112 is not equal to 84. \begin{align*}\frac{4}{7}\end{align*} and \begin{align*}\frac{12}{28}\end{align*} are not proportional.

### Review

Use the cross products to determine if the following ratios are proportional.

1. \begin{align*}\frac{1}{2} = \frac{6}{12}\end{align*}
2. \begin{align*}\frac{1}{3} = \frac{4}{12}\end{align*}
3. \begin{align*}\frac{1}{4} = \frac{3}{15}\end{align*}
4. \begin{align*}\frac{5}{6} = \frac{10}{12}\end{align*}
5. \begin{align*}\frac{3}{4} = \frac{6}{10}\end{align*}
6. \begin{align*}\frac{2}{5} = \frac{6}{15}\end{align*}
7. \begin{align*}\frac{2}{7} = \frac{4}{21}\end{align*}
8. \begin{align*}\frac{4}{7} = \frac{12}{21}\end{align*}
9. \begin{align*}\frac{7}{8} = \frac{14}{16}\end{align*}
10. \begin{align*}\frac{25}{75} = \frac{1}{3}\end{align*}
11. \begin{align*}\frac{11}{33} = \frac{1}{3}\end{align*}
12. \begin{align*}\frac{15}{33} = \frac{2}{3}\end{align*}
13. \begin{align*}\frac{18}{30} = \frac{36}{60}\end{align*}
14. \begin{align*}\frac{1}{3} = \frac{6}{12}\end{align*}
15. \begin{align*}\frac{85}{100} = \frac{43.5}{50}\end{align*}

To see the Review answers, open this PDF file and look for section 8.6.

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### Vocabulary Language: English

TermDefinition
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.