# 5.8: Changing Fractions to Decimals

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

## Introduction

*The Survey*

After the sixth grade social, the students were given a survey. The survey asked the students about their favorite events and whether or not they had a terrific time. Some of the students wished there had been more dancing, some wished there had been more games, and some wished they could have attended at all. For different reasons, 6A had 40 out of 48 students attend and 6B had 42 out of 44 students attend.

After going through the surveys, Wendy, the class secretary, has tallied the results. She is most interested in figuring out which cluster has more students who say that they had a terrific time at the social.

Here are the results.

In 6A, \begin{align*}\frac{36}{40}\end{align*} said that they had a terrific time.

In 6B, \begin{align*}\frac{35}{42}\end{align*} said that they had a terrific time.

Wendy thinks that 6A has more students who say that they had a terrific time. **Is she correct?**

**To figure out whether or not Wendy is correct, it would help to learn how to convert fractions to decimals. Once you have learned these skills, you can come back to this problem and figure out if Wendy has accurately analyzed the situation.**

*What You Will Learn*

In this lesson, you will learn the following skills:

- Write fractions as decimals.
- Write mixed numbers as decimals.
- Write fractions and mixed numbers as repeating decimals.
- Compare and Order decimals and fractions.

*Teaching Time*

I. **Write Fractions as Decimals**

In our last lesson we learned how to convert decimals to fractions. Now we are going to work the other way around.

**How do we convert a fraction to a decimal?**

**There are two ways to convert a fraction to a decimal.**

**The first way is to think in terms of place value. If we have a fraction that has ten as a denominator, we can think of that fraction as tenths. Now we can figure out how to write the decimal.**

Example

\begin{align*}\frac{6}{10}=.6\end{align*} There is one decimal place in tenths, so this decimal is accurate.

Example

\begin{align*}\frac{125}{1000}\end{align*} There are three decimal places in a thousandths decimal. There are three digits in the numerator. We can easily convert this to a decimal.

**Our answer is .125.**

**The second way is to use division. We can take the numerator of a fraction and divide it by the denominator. The denominator is the divisor and the numerator is the dividend.**

Example

\begin{align*}\frac{3}{5}\end{align*} We want to change \begin{align*}\frac{3}{5}\end{align*} to a decimal. We can do this by dividing the numerator by the denominator. We will be adding a decimal point and zero placeholders to help us with this division. Let’s take a look.

\begin{align*}& \overset{ \quad \ .6}{5 \overline{ ) {3.0 \;}}}\\ & \ \underline{-30}\\ & \quad \ \ 0\end{align*}

**Our answer is .6.**

**Now it is time for you to try a few. Convert each fraction to a decimal.**

- \begin{align*}\frac{8}{10}\end{align*}
- \begin{align*}\frac{5}{100}\end{align*}
- \begin{align*}\frac{4}{5}\end{align*}

*Take a few minutes to check your work. Did you remember the extra zero in number two?*

II. **Write Mixed Numbers as Decimals**

We can use the two methods that we used in the last section to write mixed numbers as decimals.

**How can we write a decimal from a mixed number that has a base ten denominator?**

**When we have a base ten denominator in the fraction part of a mixed number, we can think in terms of place value. Read the fraction to yourself and picture what this would look like as a decimal. You know that tenths have one decimal place, hundredths have two, thousandths have three and so on. This information will guide you in your work writing decimals.**

Let’s look at an example.

Example

\begin{align*}5\frac{3}{10}\end{align*}

The five is our whole number it goes to the left of the decimal point. Three-tenths can become .3.

**Our final answer is 5.3.**

**How do we write a decimal when we don’t have a base ten denominator?**

When this happens, we need to use division.

Example

\begin{align*}8\frac{1}{5}\end{align*}

We have the 8 as our whole number. It goes to the left of the decimal point. We divide 1 by 5 to get our decimal part of the number. Once again, we add a decimal point and a zero placeholder to divide completely.

\begin{align*}& \overset{ \quad \ .2}{5 \overline{ ) {1.0 \;}}}\\ & \ \underline{-10}\\ & \quad \ \ 0\end{align*}

**Our final answer is 8.2.**

**Try a few of these on your own.**

- \begin{align*}6\frac{13}{100}\end{align*}
- \begin{align*}15\frac{9}{10}\end{align*}
- \begin{align*}6\frac{1}{4}\end{align*}

III. **Write Fractions and Mixed Numbers as Repeating Decimals**

When we can convert a fraction by dividing the numerator by the denominator evenly to form a decimal, we call this a ** terminating decimal**. The word “terminate” means to end. All of the fractions we have been working with are terminating decimals. Let’s look at an example.

Example

\begin{align*}7\frac{1}{4}\end{align*}

Here the 7 is our whole number and so it is placed to the left of the decimal point. We divide 1 by 4 to get the decimal part.

\begin{align*}& \overset{ \quad \ .25}{4 \overline{ ) {1.00 \;}}}\\ & \ \ \underline{-8}\\ & \quad \ \ 20\\ & \quad \underline{-20}\\ & \qquad \ 0 \end{align*}

**This is a terminating decimal. It is called that because once you added the decimal point and the zero placeholders, you were able to divide the dividend by the divisor evenly.**

**What do we call a decimal that is NOT a terminating decimal?**

A decimal that does not end and repeats the same number over and over again is called a ** repeating decimal.** You know that you have a repeating decimal if when you divide the numerator by the denominator, if you keep ending up with the same number.

Here is an example.

Example

Convert \begin{align*}\frac{2}{3}\end{align*} to a decimal.

First, this does not have a base ten denominator so we will divide the numerator by the denominator.

\begin{align*}& \overset{ \quad \ .666}{3 \overline{ ) {2.000 \;}}}\\ & \ \underline{-18}\\ & \quad \ \ 20\\ & \quad \underline{-18}\\ & \qquad \ 20 \\ & \quad \ \ \underline{-18}\\ & \qquad \quad 2 \end{align*}

*Look at what happens as we divide!!! The same remainder keeps showing up and our quotient becomes a series of 6’s. It doesn’t matter if we keep adding zeros forever, our decimal will always repeat.*

**When you have a decimal that is a repeating decimal, we can add a line over the last digit in the quotient. This is a clue that the decimal repeats.**

**Our answer is** \begin{align*}.66\bar{6}\end{align*}.

**Divide these fractions and see if you end up with any repeating decimals.**

- \begin{align*}\frac{1}{3}\end{align*}
- \begin{align*}\frac{1}{8}\end{align*}
- \begin{align*}5\frac{1}{2}\end{align*}

*Take a minute to check your work with a peer.*

## Real Life Example Completed

*The Survey*

**Now you are ready to help Wendy with her comparing and her survey results.**

**Here is the problem once again.**

After the sixth grade social, the students were given a survey. The survey asked the students about their favorite events and whether or not they had a terrific time. Some of the students wished there had been more dancing, some wished there had been more games, and some wished they could have attended at all. For different reasons, 6A had 40 out of 48 students attend and 6B had 42 out of 44 students attend.

After going through the surveys, Wendy, the class secretary, has tallied the results. She is most interested in figuring out which cluster has more students who say that they had a terrific time at the social.

Here are the results.

In 6A, \begin{align*}\frac{36}{40}\end{align*} said that they had a terrific time.

In 6B, \begin{align*}\frac{35}{42}\end{align*} said that they had a terrific time.

Wendy thinks that 6A has more students who say that they had a terrific time. **Is she correct?**

**First, take a minute to underline all of the important information.**

**If you look at the survey results for 6A and 6B you will see that they both have different denominators. Their denominators aren’t base ten denominators either.**

**We need to convert both fractions into decimals to be able to compare them. The easiest way to do this is to divide using a decimal point and zero placeholders.**

\begin{align*}& 6A = \overset{ \qquad \ .9}{40 \overline{ ) {36.0 \;}}}\\ & \qquad \quad \ \underline{-360}\\ & \qquad \qquad \quad 0\end{align*}

**6A has .9 or** \begin{align*}\frac{9}{10}\end{align*} **of the students who say that they had a terrific time.**

\begin{align*}& 6B = \overset{ \qquad \ .833}{42 \overline{ ) {35.000 \;}}}\\ & \qquad \quad \ \underline{-336\;}\\ & \qquad \qquad \ \ 140\\ & \qquad \qquad \underline{-126\;}\\ & \qquad \qquad \quad \ 140 \end{align*}

**6B has a repeating decimal of** \begin{align*}.8\bar{3}\end{align*}.

**Now we can compare the two decimals.**

**.9 > .83**

**Wendy was correct. 6A does have more students who say that they had a terrific time!!**

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- Terminating Decimal
- decimal that can be found dividing a numerator and denominator and by adding a decimal point and zero placeholders.

- Repeating Decimal
- a decimal where the digits in the quotient repeat themselves, can be indicated by putting a small line over the second repeating digit.

## Technology Integration

Khan Academy Converting Fractions to Decimals

James Sousa Fractions to Decimals

James Sousa Example of Fractions to a Terminating Decimal

James Sousa Another Example of Fractions to a Terminating Decimal

James Sousa Example of Fractions to a Repeating Decimal

Other Videos:

http://www.mathplayground.com/howto_fractions_decimals.html – This is a how-to video on how to convert fractions into decimals.

## Time to Practice

Directions: Write the following fractions as decimals.

- \begin{align*}\frac{3}{10}\end{align*}
- \begin{align*}\frac{23}{100}\end{align*}
- \begin{align*}\frac{9}{100}\end{align*}
- \begin{align*}\frac{8}{10}\end{align*}
- \begin{align*}\frac{182}{1000}\end{align*}
- \begin{align*}\frac{25}{100}\end{align*}
- \begin{align*}\frac{6}{10}\end{align*}
- \begin{align*}\frac{125}{1000}\end{align*}
- \begin{align*}\frac{1}{10}\end{align*}
- \begin{align*}\frac{2}{100}\end{align*}
- \begin{align*}\frac{1}{2}\end{align*}
- \begin{align*}\frac{1}{4}\end{align*}
- \begin{align*}\frac{3}{4}\end{align*}
- \begin{align*}\frac{3}{6}\end{align*}
- \begin{align*}\frac{3}{5}\end{align*}
- \begin{align*}4\frac{1}{2}\end{align*}
- \begin{align*}7\frac{1}{3}\end{align*}
- \begin{align*}5\frac{2}{10}\end{align*}
- \begin{align*}9\frac{1}{8}\end{align*}
- \begin{align*}10\frac{2}{100}\end{align*}

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