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# Ratios in Simplest Form

## Simplify ratios using greatest common factors.

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Ratios in Simplest Form
Credit: Texas A&M University-Commerce Marketing Communication Photography
Source: https://www.flickr.com/photos/tamuc/14404498081

Grace is working on an assignment. She has to survey 50 people and ask if they are either right-handed or left-handed. Out of the 50 people she surveyed, 5 were left-handed and 45 were right-handed. How can Grace use this information to describe the results of her survey?

In this concept, you will learn how to simplify ratios and then compare and draw conclusions.

### Simplifying Ratios

A ratio is the comparison of two quantities. Ratios can involve large quantities that may not represent a clear comparison. Simplify ratios to make them easier to evaluate. Since ratios can also be written as fractions, you can simplify ratios the same way you simplify fractions.

Let’s look at the ratio of left-handed people to right-handed people. The ratio of left-handed people to right handed people is 5 to 45. First, write the ratio using the fraction notation.

\begin{align*}\frac{5}{45}\end{align*}

Then, find the greatest common factor (GCF) to find the simplest form of the ratio. The GCF is the largest factor share by both numbers. The GCF of 5 and 45 is 5.

Next, divide the numerator and the denominator by the GCF.

\begin{align*}\frac{5 \div 5}{45 \div 5}=\frac{1}{9}\end{align*}

The simplest form of the ratio \begin{align*}\frac{5}{45}\end{align*} is \begin{align*}\frac{1}{9}\end{align*}, which can also be written as 1 to 9 or 1 : 9.

There is one left-handed person for every nine right-handed people.

Remember, when you simplify a ratio, the value of the ratio does not change. Therefore, a ratio and its simplest form are equivalent ratios.

\begin{align*}\frac{5}{45}=\frac{1}{9}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Grace’s survey.

Of the 50 people she surveyed, 5 were left-handed and 45 were right-handed. Use a different ratio. Simplify the ratio and draw a conclusion.

First, decide which ratio to use and write it as a fraction. Let’s use the ratio of left-handed people to the total number of people surveyed.

\begin{align*}\frac{5}{50}\end{align*}

Next, find the GCF. The GCF of 5 and 50 is 5.

Then, divide the numerator and the denominator by 5.

\begin{align*}\frac{5 \div 5}{50 \div 5}=\frac{1}{10}\end{align*}

The simplest form of \begin{align*}\frac{5}{50}\end{align*} is \begin{align*}\frac{1}{10}\end{align*}.

There is one left-handed person for every ten people surveyed.

#### Example 2

Simplify the following ratio: \begin{align*}\frac{12}{18}\end{align*}. Write the simplified ratio as a fraction.

First, find the GCF of 12 and 18. The GCF is 6.

Next, divide the numerator and the denominator by 6.

\begin{align*}\frac{12 \div 6}{18 \div 6}=\frac{2}{3}\end{align*}

The simplest form of \begin{align*}\frac{12}{18}\end{align*} is \begin{align*}\frac{2}{3}\end{align*}.

#### Example 3

Simplify the following ratio: \begin{align*}\frac{2}{10}\end{align*}. Write the simplified ratio as a fraction.

First, find the GCF of 2 and 10. The GCF is 2.

Next, divide the numerator and denominator by 2.

\begin{align*}\frac{2 \div 2}{10 \div 2}=\frac{1}{5}\end{align*}

The simplest form of \begin{align*}\frac{2}{10}\end{align*} is \begin{align*}\frac{1}{5}\end{align*}.

#### Example 4

Simplify the following ratio: \begin{align*}6 \text{ to } 8\end{align*}. Write the simplified ratio as a fraction.

First, find the GCF of 6 and 8. The GCF is 2.

Next, divide the both numbers by 2.

\begin{align*}\begin{array}{rcl} 6 \div 2 & = & 3\\ 8 \div 2 & = & 4 \end{array}\end{align*}

Then, write 3 to 4 as a fraction.

\begin{align*}\frac{3}{4}\end{align*}

The simplest form of 6 to 8 is 3 to 4 or \begin{align*}\frac{3}{4}\end{align*}.

#### Example 5

Simplify the following ratio: \begin{align*}5: 20\end{align*}. Write the simplified ratio as a fraction.

First, find the GCF for 5 and 20. The GCF is 5.

Next, divide both numbers by 5.

\begin{align*}\begin{array}{rcl} 5 \div 5 & = & 1\\ 20 \div 5 & = & 4 \end{array}\end{align*}

Then, write 1 : 4 as a fraction.

\begin{align*}\frac{1}{4}\end{align*}

The simplest form of 5 : 20 is 1 : 4 or \begin{align*}\frac{1}{4}\end{align*}.

### Review

Find the simplest form for each ratio. Write your answer as a fraction.

1. 2 to 4
2. 3 : 6
3. 5 to 15
4. 2 to 30
5. 10 to 15
6. \begin{align*}\frac{4}{6}\end{align*}
7. 3 : 9
8. 6 : 8
9. \begin{align*}\frac{2}{8}\end{align*}
10. \begin{align*}\frac{4}{16}\end{align*}
11. 10 to 12
12. 7 : 21
13. 12 : 24
14. 25 to 75
15. \begin{align*}\frac{27}{30}\end{align*}
16. \begin{align*}\frac{48}{60}\end{align*}
17. \begin{align*}\frac{18}{80}\end{align*}

To see the Review answers, open this PDF file and look for section 8.2.

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### Vocabulary Language: English

TermDefinition
Equivalent Equivalent means equal in value or meaning.
Greatest Common Factor The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.