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### Course: 8th grade (Illustrative Mathematics)>Unit 8

Lesson 15: Lesson 14: Decimal representations of rational numbers

# Writing fractions as repeating decimals review

Review converting fractions to repeating decimals, and then try some practice problems.

## Writing fractions as decimals

To convert a fraction to a decimal, we divide the numerator by the denominator.
Example:start fraction, 2, divided by, 5, end fraction
start fraction, start color #11accd, 2, end color #11accd, divided by, start color #1fab54, 5, end color #1fab54, end fraction, equals, start color #11accd, 2, end color #11accd, divided by, start color #1fab54, 5, end color #1fab54
\begin{aligned} &~~ \, 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{aligned}
start fraction, start color #11accd, 2, end color #11accd, divided by, start color #1fab54, 5, end color #1fab54, end fraction, equals, 0, point, 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:start fraction, 4, divided by, 9, end fraction
start fraction, start color #11accd, 4, end color #11accd, divided by, start color #1fab54, 9, end color #1fab54, end fraction, equals, start color #11accd, 4, end color #11accd, divided by, start color #1fab54, 9, end color #1fab54
\begin{aligned} &~~ \, 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{aligned}
No matter how long we divide, the 4 will continue to repeat in our quotient.
start fraction, start color #11accd, 4, end color #11accd, divided by, start color #1fab54, 9, end color #1fab54, end fraction, equals, 0, point, start overline, 4, end overline

## Practice

Problem 1
• Current
Select the decimal that is equivalent to start fraction, 2, divided by, 3, end fraction.

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

## Want to join the conversation?

• why does math have to be hard like life is hard already
• Math is the reason why life is hard to begin with lol
• UH I am still a bit confused like how would we solve 0.36 but only the 6 is repeating
• Let x = 0.3666....

Only 1 digit (the 6) repeats, so multiply both sides by 10^1=10 to get

10x = 3.6666...

Subtract the first equation from the second equation to cancel out repeating parts, to get

9x = 3.3

Divide both sides by 9 to get

x = 3.3/9 .

We can simplify 3.3/9 by first multiplying top and bottom by 10 (getting rid of the decimal) to get 33/90, then dividing top and bottom by 3 to get 11/30.

Have a blessed, wonderful day!
• why does the math have to be so hard
• Math makes life hard.
• Thanks to Corona Virus, My teacher posted 402 assignments on here
• i now right
(1 vote)
• my car handle hit the car door
• I am a bit confused when 2 numbers repeat, when I do it it sometimes work and sometimes not… idk