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Proportions Using Cross Products

Cross-multiply to solve proportions with one variable

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Practice Proportions Using Cross Products
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Write and Solve Proportions by Using Cross-Products

Josh is very excited about his book on Mount Everest. He took the book to school and has been reading it every chance he can. In fact, he finished his work in math class so quickly that Ms. Henje made him check his work to be sure that it was accurate. Josh was very excited that it was accurate.

Josh looked at the clock. He still had 18 minutes left to read. Josh opened the book and read about Sir Edmund Hillary and Tenzing Norgay and their first ascent up the mountain on May 29, 1953. He was so engrossed in his reading that he did not even hear the bell ring.

“Time to go,” his friend Evan said tipping the book a little as he went by.

Josh looked up at the clock and then down at this book. He had read 10 pages in 18 minutes. Josh was excited. He picked up the book and thought about silent reading time. He had read 10 pages in 18 minutes so he was sure that he would read more pages during silent reading time since that was a 30 minutes block of time. Josh began to think about the number of pages that he would read during silent reading time.

Is this true? If Josh reads during silent reading time at the same rate that he did during math class, how many pages will he read? To figure this out, you will need to know how to write a proportion and solve it. This Concept will teach you all that you need to know so you can apply what you learn to this dilemma.

Guidance

A proportion is created when two ratios are equal. Sometimes, you will know three parts of a proportion and there will be one missing part. When this happens, you will need to solve a proportion. Let's look at how to solve proportions.

A way of solving a proportion is called cross-multiplying, and it involves algebra. Here is the rule.

If \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}ad = cb\end{align*}.

This is also called “the product of the means is equal to the product of the extremes.” The values in the \begin{align*}b\end{align*} and \begin{align*}c\end{align*} positions are called the means, and the values in the \begin{align*}a\end{align*} and \begin{align*}d\end{align*} positions are called the extremes.

However, you can just think of multiplying the values that are diagonal to each other, making an \begin{align*}X\end{align*}. After cross-multiplying, you can use algebra to solve for the variable.

Let’s apply this information now.

\begin{align*}\frac{x}{5} = \frac{9}{10}\end{align*}

With this problem, we have been given a proportion that needs solving. To solve it, we can cross-multiply.

\begin{align*}10(x) = 10x\end{align*}

\begin{align*}9(5) = 45\end{align*}

Next, we can solve using algebra. We divide 45 by 10 to find the value of the variable.

Here is another one.

\begin{align*}\frac{4}{5} = \frac{16}{x}\end{align*}

This proportion is written differently because the variable is in a different location. However, we can still solve it by using cross-products.

Solve each proportion by using cross-products.

Example A

\begin{align*}\frac{x}{9} = \frac{18}{27}\end{align*}

Solution:  \begin{align*}x = 6\end{align*}

Example B

\begin{align*}\frac{3}{7} = \frac{33}{y}\end{align*}

Solution:  \begin{align*}y = 77\end{align*}

Example C

\begin{align*}\frac{x}{2} = \frac{49.5}{99}\end{align*}

Solution:  \begin{align*}x = 1\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

First, we need to write a proportion to show the comparison between time and pages read.

Let’s begin with math class.

Josh read 10 pages in 18 minutes. Let’s write the first ratio.

Next, Josh will read for 30 minutes during silent reading time. We need to figure out the number of pages, so that is our unknown.

Here is the proportion.

We can cross multiply and solve.

Josh will read about \begin{align*}16 \frac{1}{2}\end{align*} pages during silent reading time.

Vocabulary

Ratio
a comparison between two quantities. Ratios can be written in fraction form, with a colon or by using the word “to”.
Equivalent
means equal.
Proportion
formed when two ratios are equivalent. We compare two ratios, they are equal and so they form a proportion.

Guided Practice

Here is one for you to try on your own.

The ratio of apples to bananas at a store is 3 to 8. If there are 90 apples, how many bananas are there?

Solution

Set up a proportion. Be sure that your units match up. Here apples are in the numerator position and bananas are in the denominator position. Be sure that this stays consistent throughout your work.

\begin{align*}\frac{apples}{bananas} : \frac{3}{8} = \frac{90}{x}\end{align*}

Now cross - multiply and divide.

\begin{align*}3x = 720\end{align*}

\begin{align*}x = 240\end{align*}

There are 240 bananas at the store.

Practice

Directions: Solve each proportion by using cross – multiplying with algebra. You may round to the nearest tenth when necessary.

1. \begin{align*}\frac{3}{5} = \frac{y}{2.5}\end{align*}
2. \begin{align*}\frac{6}{7} = \frac{2.5}{y}\end{align*}
3. \begin{align*}\frac{4}{5} = \frac{2}{x}\end{align*}
4. \begin{align*}\frac{9}{11} = \frac{14}{x}\end{align*}
5. \begin{align*}\frac{2}{3} = \frac{5}{y}\end{align*}
6. \begin{align*}\frac{12}{3} = \frac{4}{y}\end{align*}
7. \begin{align*}\frac{22}{40} = \frac{11}{x}\end{align*}
8. \begin{align*}\frac{60}{x} = \frac{5}{10}\end{align*}
9. \begin{align*}\frac{12}{50} = \frac{3}{y}\end{align*}
10. \begin{align*}\frac{42}{36} = \frac{7}{y}\end{align*}
11. \begin{align*}\frac{56}{63} = \frac{x}{9}\end{align*}
12. \begin{align*}\frac{120}{130} = \frac{1.2}{y}\end{align*}

Directions: Solve each problem.

1. The ratio of fiction to nonfiction books at a library is 5 to 3. If there are 480 nonfiction books, write a proportion that could be used to find \begin{align*}f\end{align*}, the number of fiction books.
2. The ratio of cherry trees to apple trees at an orchard is 4 to 9. If there are 184 cherry trees, write a proportion that could be used to find \begin{align*}a\end{align*}, the number of apple trees.
3. The ratio of cars to SUVs in a parking lot is 10 to 7. If there are 84 SUVs, write a proportion that could be used to find \begin{align*}c\end{align*}, the number of cars in the lot.

Vocabulary Language: English

Cross Products

Cross Products

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

Equivalent

Equivalent means equal in value or meaning.
Proportion

Proportion

A proportion is an equation that shows two equivalent ratios.