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# Fractions as Decimals

## Dividing the denominator into the numerator

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Fractions as Decimals
Erik is making Italian bread to go with his mom's lasagna for dinner tonight. Instead of using measuring cups to measure his ingredients by volume, he is using a scale to measure his ingredients by weight. Erik’s recipe calls for  1058\begin{align*}10 \frac{5}{8}\end{align*} ounces of water. The numbers on Erik’s scale are shown in decimal form. How can Erik figure out what number his scale should show when he has added the correct amount of water?

In this concept, you will learn how to write fractions and mixed numbers as terminating decimals.

### Writing Fractions and Mixed Numbers as Decimals

Fractions and decimals are both ways of describing numbers that are not whole numbers. Every fraction is equivalent to some decimal. To write a fraction as a decimal, remember that a fraction is really just a division expression. For example, 34\begin{align*}\frac{3}{4}\end{align*} is the same as 3÷4\begin{align*}3 \div 4\end{align*}.

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

1. Rewrite your fraction as a division problem.
2. Divide using long division. Add a decimal point and zeros to the dividend as needed.

Here is an example.

Convert 34\begin{align*}\frac{3}{4}\end{align*} to a decimal.

First, write 34\begin{align*}\frac{3}{4}\end{align*} as a division problem.

34\begin{align*}\frac{3}{4}\end{align*} is the same as 3÷4\begin{align*}3 \div 4\end{align*}.

Next, use long division to divide. You will need to add a decimal point and zeros to the 3.

4)3.00¯¯¯¯¯¯¯¯¯¯¯    0.7528   20200\begin{align*}& \overset{ \ \ \ \ 0.75}{4 \overline{ ) {3.00 }}}\\ & \underline{\; -28\;\;\;}\\ & \quad \ \ \ 20 \\ & \quad \underline{ -20} \\ & \qquad 0\end{align*}

The answer is 34=0.75\begin{align*}\frac{3}{4}=0.75\end{align*}.

You can also convert mixed numbers to decimals. Here are the steps for writing mixed numbers as decimals.

1. Convert the fractional part of the mixed number to a decimal using long division.
2. Add the whole number part of the mixed number to the result from step 1.

Here is an example.

Convert 312\begin{align*}3 \frac{1}{2}\end{align*} to a decimal.

First, set aside the 3. 12\begin{align*}\frac{1}{2}\end{align*} is the same as 1÷2\begin{align*}1 \div 2\end{align*}. Convert the 12\begin{align*}\frac{1}{2}\end{align*} to a decimal using long division.

2)1.0¯¯¯¯¯¯¯¯¯¯   0.510  0\begin{align*}& \overset{ \ \ \ 0.5}{ 2 \overline{ ) {1.0 \;}}}\\ & \underline{\; -10}\\ & \ \quad \ 0\end{align*}

\begin{align*}\frac{1}{2}=0.5\end{align*}

Next, add the 3 from the original mixed number.

\begin{align*}3.5\end{align*}

The answer is \begin{align*}3 \frac{1}{2}=3.5\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Erik and his Italian bread.

He needs \begin{align*}10 \frac{5}{8}\end{align*} ounces of water to make his bread dough, but his scale only shows numbers in decimal form. Erik needs to figure out what number his scale should show when he has added the correct amount of water.

Erik needs to convert \begin{align*}10 \frac{5}{8}\end{align*} to a decimal.

First, he should set aside the 10. \begin{align*}\frac{5}{8}=5 \div 8\end{align*}. He needs to convert the \begin{align*}\frac{5}{8}\end{align*} to a decimal using long division.

\begin{align*}& \overset{ \ \ \ 0.625}{8 \overline{ ) {5.000 }}}\\ & \underline{\; -48\;\;\;}\\ & \quad \ \ \ 20 \\ & \quad \underline{ -16 \; \;} \\ & \quad \ \ \ \ \ 40 \\ & \quad \ \ \underline{ -40 \; \;} \\ & \qquad \ \ \ 0\end{align*}

\begin{align*}\frac{5}{8}=0.625\end{align*}

Next, he should add the 10 from the original mixed number.

\begin{align*}10.625\end{align*}

The answer is Erik should look for his scale to show 10.625 when he has added the correct amount of water.

#### Example 2

Convert \begin{align*}4 \frac{3}{5}\end{align*} to a decimal.

First, set aside the 4. \begin{align*}\frac{3}{5}\end{align*} is the same as \begin{align*}3 \div 5\end{align*}. Convert the \begin{align*}\frac{3}{5}\end{align*} to a decimal using long division.

\begin{align*}& \overset{ \ \ \ 0.6}{ 5 \overline{ ) {3.0 }}}\\ & \underline{\; -30}\\ & \ \quad \ 0\end{align*}

\begin{align*}\frac{3}{5}=0.6\end{align*}

Next, add the 4 from the original mixed number.

\begin{align*}4.6\end{align*}

The answer is \begin{align*}4 \frac{3}{5}=4.6\end{align*}.

#### Example 3

Convert \begin{align*}\frac{1}{5}\end{align*} to a decimal.

First, write \begin{align*}\frac{1}{5}\end{align*} as a division problem.

\begin{align*}\frac{1}{5}\end{align*} is the same as \begin{align*}1 \div 5\end{align*}.

Next, use long division to divide. You will need to add a decimal point and a zero to the 1.

\begin{align*}& \overset{ \ \ \ 0.2}{5 \overline{ ) {1.0 }}} \\ & \ \underline{-10}\\ & \ \ \quad 0\end{align*}

The answer is \begin{align*}\frac{1}{5}=0.2\end{align*}.

#### Example 4

Convert \begin{align*}\frac{3}{6}\end{align*} to a decimal.

First, write \begin{align*}\frac{3}{6}\end{align*} as a division problem.

\begin{align*}\frac{3}{6}\end{align*} is the same as \begin{align*}3 \div 6\end{align*}.

Next, use long division to divide. You will need to add a decimal point and a zero to the 3.

\begin{align*}& \overset{ \ \ \ \ 0.5}{ 6 \overline{ ) {3.0 }}}\\ & \underline{\; -30}\\ & \ \quad \ 0\end{align*}

The answer is \begin{align*}\frac{3}{6}=0.5\end{align*}.

#### Example 5

Convert \begin{align*}4 \frac{4}{5}\end{align*} to a decimal.

First, set aside the 4. \begin{align*}\frac{4}{5}=4 \div 5\end{align*}. Convert the \begin{align*}\frac{4}{5}\end{align*} to a decimal using long division.

\begin{align*}& \overset{ \ \ \ \ 0.8}{ 5 \overline{ ) {4.0 }}}\\ & \underline{\; -40}\\ & \ \quad \ 0\end{align*}

\begin{align*}\frac{4}{5}=0.8\end{align*}

Next, add the 4 from the original mixed number.

\begin{align*}4.8\end{align*}

The answer is \begin{align*}4 \frac{4}{5}=4.8\end{align*}.

### Review

Convert each fraction or mixed number to a decimal.

1. \begin{align*}\frac{3}{5}\end{align*}
2. \begin{align*}\frac{1}{2}\end{align*}
3. \begin{align*}\frac{4}{5}\end{align*}
4. \begin{align*}\frac{3}{4}\end{align*}
5. \begin{align*}\frac{3}{8}\end{align*}
6. \begin{align*}\frac{7}{8}\end{align*}
7. \begin{align*}2\frac{1}{4}\end{align*}
8. \begin{align*}5\frac{2}{5}\end{align*}
9. \begin{align*}6\frac{4}{8}\end{align*}
10. \begin{align*}11\frac{5}{8}\end{align*}
11. \begin{align*}10\frac{1}{2}\end{align*}
12. \begin{align*}18\frac{3}{4}\end{align*}
13. \begin{align*}12\frac{4}{5}\end{align*}
14. \begin{align*}19\frac{3}{4}\end{align*}
15. \begin{align*}22\frac{1}{4}\end{align*}

### Vocabulary Language: English

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.

Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.

Place Value

The value of given a digit in a multi-digit number that is indicated by the place or position of the digit.

Terminating Decimal

A terminating decimal is a decimal number that ends. The decimal number 0.25 is an example of a terminating decimal.