Repeating decimals · EduSite Academy

▶ Converting a fraction to a repeating decimal

To convert a fraction like 19/27 into a decimal, you should divide 27 into 19, and use the remainder to figure out the decimal. The decimal begins 0.703703..., and the notation for a repeating decimal like this is to write the numbers that repeat and then put a line above them. The first six digits of the decimal should be included in the answer.

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▶ Converting repeating decimals to fractions (part 1 of 2)

The process of converting a repeating decimal to a fraction can be broken down into a few easy steps. To start, set the decimal equal to a variable. Multiply the decimal by 10 and subtract the original decimal from it. Finally, divide both sides by 9 to obtain the fractional form of the decimal. For example, 0.7 repeating would be 7/9, and 1.2 repeating would be 11/9.

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▶ Converting repeating decimals to fractions (part 2 of 2)

Repeated decimals can be converted into fractions by shifting the decimal to the right and subtracting the decimals. To do this, multiply the number by 10 to the second power, then subtract. For example, 0.363636 repeating is 4/11 and 0.7141414 repeating is 707/990. Another example is 3.257257257 repeating, which is 3254/999. This calculation can be done in the head or by borrowing. After the subtraction, the numerator and denominator can be reduced and the fraction can be simplified.

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📄 Writing repeating decimals as fractions review

##Writing decimals as fractions

To convert a decimal to a fraction, we write the decimal number as a numerator, and we write its place value as the denominator.

Example 1: $0.07$

$0.0\blueD7$ is $\blueD7$ $\text{\greenD{hundredths}}$. So, we write $\blueD7$ over $\greenD{100}$.

$0.07=\dfrac{\blueD7}{\greenD{100}}$

##But what about repeating decimals?

Let's look at an example.

Rewrite $0.\overline{7}$ as a simplified fraction.

Let $x$ equal the decimal:

$\large{x = 0.7777...} $

Set up a second equation such that the digits after the decimal point are identical:

$\large{\begin{align*} 10x &= 7.7777...\ x &= 0.7777... \end{align*}} $

Subtract the two equations:

$\large{9x = 7}$

Solve for $x$:

$\large{ x = \dfrac{7}{9}} $

Remember from the first step that $x$ is equal to our repeating decimal, so:

$\large{0.\overline{7}=\dfrac79}$

Want to learn more about writing repeating decimals as fractions? Check out this video.

##Practice

[[☃ graded-group-set 1]]

Want to try more problems like this? Check out this exercise.

📄 Writing fractions as repeating decimals review

##Writing fractions as decimals

To convert a fraction to a decimal, we divide the numerator by the denominator.

Example:$\dfrac25$

$\dfrac\blueD2\greenD5 =\blueD2\div\greenD5$

$\begin{align}
&~~ , 0.{4} \
\greenD5&\overline{\Big)\blueD2.0} \phantom{0000000}& \
&\underline{, \mbox{-}0},,,{!\downarrow}&\
&\phantom{0}{2,,0}& \
&\underline{, \mbox{-}2,,0}&\
&\phantom{00}{,,0}&\
\end{align}$

$\dfrac\blueD2\greenD5 =0.4$

##Writing fractions as repeating decimals

However, it does not always work out that nicely.

Sometimes, when we divide the numerator by the denominator, it becomes clear that the digits will keep repeating. In this case, we write the answer as a repeating decimal.

In our answer, we write a bar over the repeating digits to show that they repeat.

Example:$\dfrac49$

$\dfrac\blueD4\greenD9 =\blueD4\div\greenD9$

$\begin{align}
&~~ , 0.{4444} \
\greenD9&\overline{\Big)\blueD4.0000} \phantom{0000000}& \
&\underline{, \mbox{-}0},,,{!\downarrow}&\
&\phantom{0}{4,,0}& \
&\underline{, \mbox{-}3,,6}{,,!\downarrow}&&\
&\phantom{00}{,,4,0}&\
&\underline{,,,,,, \mbox{-}3,6}{,,!\downarrow},,\\
&\phantom{0000}{4,0}& \
&\underline{,,,,,,,,,, \mbox{-}3,6}&\
&\phantom{000,,0}{,,4}&\
\end{align}$

No matter how long we divide, the $4$ will continue to repeat in our quotient.

$\dfrac\blueD4\greenD9 =0.\overline4$

Want to learn more about writing fractions as repeating decimals? Check out this video.

##Practice

[[☃ graded-group-set 1]]

Want to try more problems like this? Check out this exercise.