# 5.12: Proportions with Variable in the Denominator

**At Grade**Created by: CK-12

**Practice**Proportions with Variable in the Denominator

Remember Rafael? Well, he has found a few authors that he likes and so he is very invested in reading.

In six weeks, Rafael read 8 books. If he continues to read at this same pace, how many books will he read in 9 weeks?

Here is a proportion to describe this situation.

\begin{align*}\frac{weeks}{books} = \frac{weeks}{books}\end{align*}

**This Concept will teach you how to fill in the given information and solve this problem.**

### Guidance

In an earlier Concept, you learned to use cross products to solve when the variable is in the numerator. What about when the variable is in the denominator?

You can use cross products to solve these proportions as well.

Use the cross products property of proportions to solve for \begin{align*}r \ \frac{2}{8} = \frac{7}{r}\end{align*}.

**The cross product property of proportions states that the product of the means is equal to the product of the extremes.**

The means in this proportion are 8 and 7. The extremes are 2 and \begin{align*}r\end{align*}. Find the cross products and solve for \begin{align*}r\end{align*}.

\begin{align*}\frac{2}{8} &= \frac{7}{r}\\ 8 \cdot 7 &= 2 \cdot r\\ 56 &= 2r\end{align*}

In \begin{align*}2r\end{align*}, the 2 and the \begin{align*}r\end{align*} are being multiplied. The opposite of multiplication is division. So, to find the value of \begin{align*}r\end{align*}, divide both sides of the equation by 2.

\begin{align*}56 &= 2r\\ \frac{56}{2} &= \frac{2r}{2}\\ 28 &= r\end{align*}

**The value of \begin{align*}r\end{align*} is 28.**

Use the cross products property of proportions to solve for \begin{align*}s \ \frac{4}{s} = \frac{10}{15}\end{align*}.

**The means in this proportion are \begin{align*}s\end{align*} and 10. The extremes are 4 and 15. Find the cross products and solve for \begin{align*}s\end{align*}.**

\begin{align*}\frac{4}{s} &= \frac{10}{15}\\ s \cdot 10 &= 4 \cdot 15\\ 10s &= 60\\ \frac{10s}{10} &= \frac{60}{10}\\ s &= 6\end{align*}

**The value of \begin{align*}s\end{align*} is 6.**

Now try a few on your own. Solve by using cross products.

#### Example A

\begin{align*}\frac{4}{7} = \frac{16}{x}\end{align*}

**Solution: \begin{align*}x = 28\end{align*}**

#### Example B

\begin{align*}\frac{5}{9} = \frac{35}{y}\end{align*}

**Solution:\begin{align*}y = 63\end{align*}**

#### Example C

\begin{align*}\frac{9}{a} = \frac{18}{33}\end{align*}

**Solution: \begin{align*}x = 16.5\end{align*}**

Here is the original problem once again.

In six weeks, Rafael read 8 books. If he continues to read at this same pace, how many books will he read in 9 weeks?

Here is a proportion to describe this situation.

\begin{align*}\frac{weeks}{books} = \frac{weeks}{books}\end{align*}

Now let's fill in what we know.

\begin{align*}\frac{6}{8} = {9}{x}\end{align*}

Use cross products and algebra to solve.

\begin{align*}6x = 72\end{align*}

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

**At this rate, Rafael will have read 12 books in that time.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Proportion
- two equal ratios form a proportion.

- Means
- the values of a proportion that are close to each other when written in ratio form using a colon.

- Extremes
- the values of a proportion that are farther apart from each other when written in ratio form using a colon.

- Cross Product Property of Proportions
- states that the cross products of two ratios will be equal if the two ratios form a proportion.

### Guided Practice

Here is one for you to try on your own.

Each pencil sold at the school store costs the same price. Caryn paid $1.12 for 14 pencils at the school store. Arnob also bought pencils, but he paid $0.48 for the pencils he buys. Write a proportion to represent \begin{align*}n\end{align*}, the number of pencils that Arnob bought.

**Answer**

When setting up a proportion, we must be sure to use consistent terms. One way to set up a proportion would be to write two equivalent ratios, each comparing the total price to the number of pencils.

We know that Caryn paid $1.12 for 14 pencils. So, one ratio could be this one.

\begin{align*}\frac{price}{pencils} = \frac{1.12}{14}\end{align*}

We know that Arnob paid $0.48. The number of pencils he bought is unknown, so we can use \begin{align*}n\end{align*} to represent that number.

\begin{align*}\frac{price}{pencils} = \frac{0.48}{n}\end{align*}

Since these two ratios are equivalent, we can put them together to form a proportion.

\begin{align*}\frac{1.12}{14} = \frac{0.48}{n}\end{align*}

The proportion above could be used to find \begin{align*}n\end{align*}, the number of pencils Arnob bought.

**If Jake joined the two in shopping for pencils and purchased some of the same pencils, he would also have a ratio that would be equivalent to these two ratios. The situation is the same. Everyone is purchasing pencils. The pencils cost the same amount. Therefore, we have equivalent proportions as we compare pencils purchased with pencil price.**

### Video Review

Here is a video for review.

- This is a James Sousa video on solving proportions.

### Practice

Directions: Solve each proportion using cross products and algebra.

1. \begin{align*}\frac{3}{5} = \frac{6}{x}\end{align*}

2. \begin{align*}\frac{5}{10} = \frac{15}{x}\end{align*}

3. \begin{align*}\frac{6}{7} = \frac{12}{x}\end{align*}

4. \begin{align*}\frac{6}{7} = \frac{36}{x}\end{align*}

5. \begin{align*}\frac{9}{11} = \frac{81}{x}\end{align*}

6. \begin{align*}\frac{4}{12} = \frac{16}{x}\end{align*}

7. \begin{align*}\frac{16}{17} = \frac{32}{x}\end{align*}

8. \begin{align*}\frac{14}{15} = \frac{18}{x}\end{align*}

9. \begin{align*}\frac{13}{15} = \frac{39}{x}\end{align*}

10. \begin{align*}\frac{14}{15} = \frac{7}{x}\end{align*}

11. \begin{align*}\frac{18}{19} = \frac{9}{x}\end{align*}

12. \begin{align*}\frac{22}{25} = \frac{11}{x}\end{align*}

13. \begin{align*}\frac{14}{18} = \frac{42}{x}\end{align*}

14. \begin{align*}\frac{21}{12} = \frac{42}{x}\end{align*}

15. \begin{align*}\frac{19}{23} = \frac{9.5}{x}\end{align*}

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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.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.Proportion

A proportion is an equation that shows two equivalent ratios.### Image Attributions

Here you'll learn to solve proportions with a variable in the denominator using cross products.