# 8.1: Ratios and Equivalent Ratios

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

**Practice**Equivalent Ratios

Angie wants to bake cookies for the bake sale. She looks at recipes for sugar cookies. Most of the recipes say to use 3 cups of flour, 1 cup of butter, 1 cup of sugar, and some other stuff. She goes to the store to buy flour and butter. What is the ratio of flour to butter in one batch of sugar cookies? If she wants to make 3 batches of cookies, how much flour and butter will she need?

In this concept, you will learn how to write ratios and find equivalent ratios.

### Writing Ratios and Finding Equivalent Ratios

A **ratio** is a comparison of two quantities by division. Ratios describe a part-to-part comparison or a part-to-whole comparison. Ratios are written three ways: as a fraction, with a colon, and with the word “to.” The ratio for quantity \begin{align*}a\end{align*} to quantity \begin{align*}b\end{align*} is:

\begin{align*}\begin{array}{rcl} && \ \ \ \frac{a}{b}\\ && \ a:b\\ && a \text{ to } b \end{array}\end{align*}

Look at the picture below.

There are three stars and two circles. A ratio that describes the comparison of the number of stars to the number of circles is a part-to-part ratio. A ratio that describes the number of stars to the total number of items is a part-to-whole ratio.

Let’s write a ratio for the number of stars to the number of circles three different ways. When writing a ratio, the order of the numbers is important. The first number must match the first quantity and the second number must match the second quantity.

\begin{align*}\begin{array}{rcl}
&& \ \ \ \frac{3}{2}\\
&& \ 3:2\\
&& 3 \text{ to } 2
\end{array}\end{align*}

The ratio 3 to 2 tells you that there are 3 stars for every 2 circles. Some other examples of part-to-part ratios for this picture could be ratios that describe orange objects to blue objects, blue stars to orange stars, orange circles to blue stars, and many others.

Let’s write a part-to-whole ratio that describes the number of blue objects to the total number of objects. Write the ratio three different ways.

\begin{align*}\begin{array}{rcl} && \ \ \ \frac{1}{5}\\ && \ 1:5\\ && 1 \text{ to } 5 \end{array}\end{align*}

Two ratios that have the same value are called **equivalent ratios**. To find an equivalent ratio, multiply or divide both quantities by the same number. It is the same process as finding equivalent fractions.

Let’s look at the ratio \begin{align*}\frac{3}{2}\end{align*}, the ratio of the number of stars to the number of circles. Multiply both the numerator and denominator by 2.

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

The ratio \begin{align*}\frac{6}{4}\end{align*} is equivalent to \begin{align*}\frac{3}{2}\end{align*}. Compare the picture of 3 stars and 2 circles and a picture of 6 stars and 4 circles. There 3 stars for every 2 circles in both pictures.

### Examples

#### Example 1

Earlier, you were given a problem about Angie at the store.

The recipe uses 3 cups of flour and 1 cup of butter and she wants to make 3 batches of the recipe. To find how much flour and butter she needs, Angie can use equivalent ratios.

First, write a ratio for the number of cups of flour and the number of cups of butter.

\begin{align*}\frac{3 \text{ cups of flour}}{1 \text{ cups of butter}}\end{align*}

Next, find an equivalent ratio by multiplying the numerator and denominator by 3.

\begin{align*}\frac{3 \times 3}{1 \times 3}=\frac{9}{3}\end{align*}

The equivalent ratio tells you that Angie will need 9 cups of flour and 3 cups of butter to make 3 batches of cookies.

**Use the image below to answer the following questions.**

#### Example 2

What is the ratio of total marbles to blue marbles? Find an equivalent ratio.

First, count the total number of marbles and blue marbles. There are 22 marbles total and 6 blue marbles.

Then, write the quantities as a ratio three different ways. The total is the first quantity and the number of blue marbles is the second quantity.

\begin{align*}\frac{22}{6} \qquad 22:6 \qquad 22 \text{ to }6\end{align*}

Next, find an equivalent ratio by multiplying or dividing both quantities by the same number. Use the fraction form of the ratio.

\begin{align*}\frac{22 \times 3}{6 \times 3}=\frac{66}{18}\end{align*}

The ratio of total marbles to blue marbles is \begin{align*}\frac{22}{6}\end{align*}, 22:6, or 22 to 6. An equivalent ratio of \begin{align*}\frac{22}{6}\end{align*} is \begin{align*}\frac{66}{18}\end{align*}.

#### Example 3

What is the ratio of orange marbles to blue marbles?

First, count the number of orange marbles and blue marbles. There are 4 orange marbles and 6 blue marbles.

Next, write the quantities as a ratio three different ways. The number of orange marbles is the first quantity and the number of blue marbles is the second quantity.

\begin{align*}\frac{4}{6} \qquad 4:6 \qquad 4 \text{ to }6\end{align*}

Then, find an equivalent ratio by multiplying or dividing both quantities by the same number. Use the fraction form of the ratio.

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

The ratio of orange marbles to blue marbles is \begin{align*}\frac{4}{6}\end{align*}, 4:6, or 4 to 6. An equivalent ratio of \begin{align*}\frac{4}{6}\end{align*} is \begin{align*}\frac{2}{3}\end{align*}.

#### Example 4

What is the ratio of purple marbles to total marbles?

First, count the number of purple marbles and the total number of marbles. There are 5 purple marbles and 22 total marbles.

Next, write the quantities as a ratio three different ways. The number of purple marbles is the first quantity and the total number of marbles is the second quantity.

\begin{align*}\frac{5}{22} \qquad 5:22 \qquad 5 \text{ to }22\end{align*}

Then, find an equivalent ratio by multiplying or dividing both quantities by the same number. Use the fraction form of the ratio.

\begin{align*}\frac{5 \times 2}{22 \times 2}=\frac{10}{44}\end{align*}

The ratio of purple marbles to the total marbles is \begin{align*}\frac{5}{22}\end{align*}, 5:22, or 5 to 22. An equivalent ratio of \begin{align*}\frac{5}{22}\end{align*} is \begin{align*}\frac{10}{44}\end{align*}.

#### Example 5

What is the ratio of green marbles to blue marbles?

First, count the number of green marbles and blue marbles. There are 7 green marbles and 6 blue marbles.

Next, write the quantities as a ratio three different ways. The number of green marbles is the first quantity and the number of blue marbles is the second quantity.

\begin{align*}\frac{7}{6} \qquad 7:6 \qquad 7 \text{ to }6\end{align*}

Then, find an equivalent ratio by multiplying or dividing both quantities by the same number. Use the fraction form of the ratio.

\begin{align*}\frac{7 \times 3}{6 \times 3}=\frac{21}{18}\end{align*}

The ratio of green marbles to blue marbles is \begin{align*}\frac{7}{6}\end{align*}, 7:6, or 7 to 6. An equivalent ratio of \begin{align*}\frac{7}{6}\end{align*} is \begin{align*}\frac{21}{18}\end{align*}.

### Review

Use the picture to answer the following questions. Write each ratio three ways.

- What is the ratio of hens to chicks?
- What is the ratio of green chicks to yellow chicks?
- What is the ratio of white chicks to total chicks?
- What is the ratio of green chicks to total chicks?
- What is the ratio of yellow chicks to total chicks?
- What is the ratio of green chicks to white chicks?

Use the picture to answer the following questions. Find an equivalent ratio. (Equivalent ratios will vary.)

- What is the ratio of light blue marbles to dark blue ones?
- What is the ratio of orange marbles to red marbles?
- What is the ratio of pink marbles to red marbles?
- What is the ratio of green marbles to total marbles?
- What is the ratio of yellow marbles to red marbles?
- What is the ratio of total marbles to purple marbles?
- What is the ratio of total marbles to all blue marbles?
- What is the ratio of pink marbles to total marbles?
- What is the ratio of red marbles to total marbles?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.1.

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### Image Attributions

In this concept, you will learn how to write ratios and find equivalent ratios.

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