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# Proportions Using Cross Products

## Cross-multiply to solve proportions with one variable

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Proportions Using Cross Products

Remember Manuel who was reading all of the medieval books on knights in the Recognize Proportions as a Statement of Equivalent Ratios Concept? Well after he finished reading the series, he loaned it to his friend Rafael. Rafael is enjoying the series as much as Manuel did.

In five weeks, Rafael had already finished 8 of the 12 books. It took Manuel 7.5 weeks to read all 12 books. Will Rafael and Manuel finish the series in the same amount of time? Are they reading at the same rate? How can you figure this out?

To figure this out, you will need to know how to determine if two ratios form a proportion. If the reading rate of the boys is the same, then the ratios will form a proportion.

Use this Concept to learn how to solve proportions using cross products. Then you will know how to figure out this dilemma.

### Guidance

Previously we learned that a proportion states that two ratios are equivalent. Here are two proportions.

ab=cdora:b=c:d\begin{align*}\frac{a} {b} = \frac{c} {d} \qquad \text{or} \qquad 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.

In the last lesson, you learned how to solve proportions by using proportional reasoning. We can also solve a proportion for a variable in another way. This is where the cross products property of proportions comes in.

What is the Cross Products Property of Proportions?

The Cross Products Property of Proportions states that the product of the means is equal to the product of the extremes. You can find these cross products by cross multiplying, as shown below.

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

To solve this, we can multiply the means and the extremes.

a88a=46=24

Next, we solve the equation for the missing variable. To do this, we use the inverse operation. Multiplication is in the problem, so we use division to solve it. We divide both sides by 8.

8a8a=248=3

Our answer is 3.

Solve for each variable in the numerator by using cross products.

#### Example A

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

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

#### Example B

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

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

#### Example C

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

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

Here is the original problem once again.

Remember Manuel who was reading all of the medieval books on knights? Well after he finished reading the series, he loaned it to his friend Rafael. Rafael is enjoying the series as much as Manuel did.

In five weeks, Rafael had already finished 8 of the 12 books. It took Manuel 7.5 weeks to read all 12 books. Will Rafael and Manuel finish the series in the same amount of time? Are they reading at the same rate? How can you figure this out?

Let’s write a proportion to solve this problem.

8 books5 weeks=12 books7.5 weeks\begin{align*}\frac{8 \ books}{5 \ weeks} = \frac{12 \ books}{7.5 \ weeks}\end{align*}

Next, we can use cross products to see if the two ratios form a proportion. If they do, then the two boys will finish the series in the same amount of time.

8×7.55×12=60=60

The two cross products are equal so the two ratios form a proportion. The two boys will finish the series of books in the same amount of time.

### Vocabulary

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.

The ratio of boys to girls in the school chorus is 4 to 5. There are a total of 20 boys in the chorus. How many total students are in the chorus?

The ratio given, 4 to 5, compares boys to girls. However, the question asks for the total number of students in the chorus.

One way to set up a proportion for this problem would be to write two equivalent ratios, each comparing boys to total students.

The ratio of boys to girls is 4 to 5. We can use this ratio to find the ratio of boys to total students.

boystotal=boysboys+girls=44+5=49\begin{align*}\frac{boys}{total} = \frac{boys}{boys + girls} = \frac{4}{4 + 5} = \frac{4}{9}\end{align*}

You know that there are 20 boys in the chorus. The total number of students is unknown, so represent that as x\begin{align*}x\end{align*}.

boystotal=20x\begin{align*}\frac{boys}{total} = \frac{20}{x}\end{align*}

Get those ratios equal to form a proportion. Then cross multiply to solve for x\begin{align*}x\end{align*}.

49920180180445=20x=4x=4x=4x4=x

So, there are a total of 45 students in the school chorus.

### Practice

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

1. \begin{align*}\frac{6}{10} = \frac{x} {5}\end{align*}

2. \begin{align*}\frac{2}{3} = \frac{x} {9}\end{align*}

3. \begin{align*}\frac{4}{9} = \frac{a} {45}\end{align*}

4. \begin{align*}\frac{7}{8} = \frac{a} {4}\end{align*}

5. \begin{align*}\frac{b}{8} = \frac{5} {16}\end{align*}

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

7. \begin{align*}\frac{4}{x} = \frac{8} {10}\end{align*}

8. \begin{align*}\frac{1.5}{y} = \frac{3} {9}\end{align*}

9. \begin{align*}\frac{4}{11} = \frac{c} {33}\end{align*}

10. \begin{align*}\frac{2}{6} = \frac{5} {y}\end{align*}

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

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

13. \begin{align*}\frac{5}{r} = \frac{70} {126}\end{align*}

14. \begin{align*}\frac{4}{14} = \frac{14} {k}\end{align*}

15. \begin{align*}\frac{8}{w} = \frac{6} {3}\end{align*}

16. \begin{align*}\frac{2}{5} = \frac{17} {a}\end{align*}

### Vocabulary Language: English

Cross Product Property of Proportions

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

Cross Products

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

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

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.