# Decimals as Fractions

## Fractions with 10, 100 or 1000 as denominators

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Decimals as Fractions

Liam is making his great-grandmother's famous brownies. He has the recipe that his great-grandmother wrote down many years ago on an index card. As Liam looks through the ingredients he sees that he will need 1.25 cups of sugar. He takes out the measuring cups and notices that all the measurements on the cups are written as fractions. How can Liam convert 1.25 to a mixed number so that he can figure out what measuring cups to use to measure out the sugar for the brownies?

In this concept, you will learn to write terminating decimals as fractions.

### Writing Decimals as Fractions

Decimals and fractions are different ways of expressing the same numbers. Every repeating and terminating decimal is equivalent to some fraction. To write a decimal as a fraction, you will need to think about the place value of the decimal digits.

Here are the steps for writing a terminating decimal as a fraction.

1. Determine the numerator of the fraction. The numerator of the fraction will be the digits to the right of the decimal point.
2. Determine the denominator of the fraction. The denominator of the fraction will be the place value of the last digit of the decimal. For example, if the last digit of the decimal is in the thousandths place, the denominator will be 1000.

Here is an example.

Convert 0.35 to a fraction.

First, read the decimal out loud. 0.35 is “35 hundredths”. The numerator of the fraction will be 35. The denominator of the fraction will be 100.

\begin{align*}\frac{35}{100}\end{align*}

Next, simplify your answer. Both the numerator and the denominator are divisible by 5.

\begin{align*}\frac{35 \div 5}{100 \div 5}= \frac{7}{20}\end{align*}

The answer is \begin{align*}0.35= \frac{7}{20}\end{align*}.

You can convert decimals greater than 1 to mixed numbers. Here are the steps for writing a terminating decimal as a mixed number.

1. Convert the portion of the number to the right of a decimal point to a fraction. The numerator of the fraction will be the digits to the right of the decimal point. The denominator of the fraction will be the place value of the last digit of the number.
2. Simplify the fraction.
3. Add the whole number part of the original decimal to the fraction.

Here is an example.

Convert 2.4 to a mixed number.

First, set aside the 2.

Next, read the portion of the number to the right of the decimal point out loud. 0.4 is “4 tenths”. The numerator of the fraction will be 4. The denominator of the fraction will be 10.

\begin{align*}\frac{4}{10}\end{align*}

Now, simplify your fraction. Both the numerator and the denominator are divisible by 2.

\begin{align*}\frac{4 \div 2}{10 \div 2} = \frac{2}{5}\end{align*}

Finally, add the 2 from the original mixed number.

\begin{align*}2 \frac{2}{5}\end{align*}

The answer is \begin{align*}2.4 = 2 \frac{2}{5}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Liam and his brownie recipe.

The old recipe from his great-grandmother calls for 1.25 cups of sugar. Liam needs to convert this mixed number to a fraction so that he can use his measuring cups.

First, Liam should set aside the 1.

Next, he should read the portion of the number to the right of the decimal point out loud. 0.25 is “25 hundredths”. The numerator of the fraction will be 25. The denominator of the fraction will be 100.

\begin{align*}\frac{25}{100}\end{align*}

Now, he can simplify his fraction. Both the numerator and denominator are divisible by 25.

\begin{align*}\frac{25 \div 25}{100 \div 25} = \frac{1}{4}\end{align*}

Finally, he can add the 1 from the original mixed number.

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

The answer is Liam will need \begin{align*}1 \frac{1}{4}\end{align*} cups of sugar. He can use the 1 cup and the \begin{align*}\frac{1}{4}\end{align*} cup measuring cups to measure out the sugar.

#### Example 2

Convert 5.25 to a mixed number.

First, set aside the 5.

Next, read the portion of the number to the right of the decimal point out loud. 0.25 is “25 hundredths”. The numerator of the fraction will be 25. The denominator of the fraction will be 100.

\begin{align*}\frac{25}{100}\end{align*}

Now, simplify your fraction. Both the numerator and the denominator are divisible by 25.

\begin{align*}\frac{25 \div25}{100 \div 25} = \frac{1}{4}\end{align*}

Finally, add the 5 from the original mixed number.

\begin{align*}5 \frac{1}{4}\end{align*}

The answer is \begin{align*}5.25 = 5 \frac{1}{4}\end{align*}.

#### Example 3

Convert 0.5 to a fraction.

First, read the decimal out loud. 0.5 is “5 tenths”. The numerator of the fraction will be 5. The denominator of the fraction will be 10.

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

Next, simplify your answer. Both the numerator and the denominator are divisible by 5.

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

The answer is \begin{align*}0.5 = \frac{1}{2}\end{align*}.

#### Example 4

Convert 0.67 to a fraction.

First, read the decimal out loud. 0.67 is “67 hundredths”. The numerator of the fraction will be 67. The denominator of the fraction will be 100.

\begin{align*}\frac{67}{100}\end{align*}

This fraction is already in simplest form.

The answer is \begin{align*}0.67 = \frac{67}{100}\end{align*}.

#### Example 5

Convert 3.21 to a mixed number.

First, set aside the 3.

Next, read the portion of the number to the right of the decimal point out loud. 0.21 is “21 hundredths”. The numerator of the fraction will be 21. The denominator of the fraction will be 100.

\begin{align*}\frac{21}{100}\end{align*}

This fraction is already in simplest form. Now, add the 3 from the original mixed number.

\begin{align*}3 \frac{21}{100}\end{align*}

The answer is \begin{align*}3.21=3 \frac{21}{100}\end{align*}.

### Review

Write each decimal as a fraction or mixed number in simplest form.

1. 0.3
2. 0.4
3. 0.2
4. 0.53
5. 0.55
6. 0.08
7. 0.25
8. 0.23
9. 0.876
10. 0.512
11. 74.34
12. 3.88
13. 5.6
14. 12.8
15. 23.06
16. 0.987

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### Vocabulary Language: English

TermDefinition
Decimal In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).
fraction A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
Mixed Number A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.