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Writing repeating decimals as fractions review

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

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, point, 07
0, point, 0, start color #11accd, 7, end color #11accd is start color #11accd, 7, end color #11accd start text, start color #1fab54, h, u, n, d, r, e, d, t, h, s, end color #1fab54, end text. So, we write start color #11accd, 7, end color #11accd over start color #1fab54, 100, end color #1fab54.
0, point, 07, equals, start fraction, start color #11accd, 7, end color #11accd, divided by, start color #1fab54, 100, end color #1fab54, end fraction

But what about repeating decimals?

Let's look at an example.
Rewrite 0, point, start overline, 7, end overline as a simplified fraction.
Let x equal the decimal:
x, equals, 0, point, 7777, point, point, point
Set up a second equation such that the digits after the decimal point are identical:
10x=7.7777...x=0.7777...\large{\begin{aligned} 10x &= 7.7777...\\ x &= 0.7777... \end{aligned}}
Subtract the two equations:
9, x, equals, 7
Solve for x:
x, equals, start fraction, 7, divided by, 9, end fraction
Remember from the first step that x is equal to our repeating decimal, so:
0, point, start overline, 7, end overline, equals, start fraction, 7, divided by, 9, end fraction
Want to learn more about writing repeating decimals as fractions? Check out this video.

Practice

Problem 1
Rewrite as a simplified fraction.
0, point, start overline, 2, end overline, equals, question mark
  • Your answer should be
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
  • a mixed number, like 1, space, 3, slash, 4

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

Want to join the conversation?

  • starky seedling style avatar for user abbi.campbell
    what could you do if you had 0.2 repted
    (13 votes)
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  • blobby green style avatar for user mackenziebowers
    How do we repeating decimals to fractions? I have a problem find the repeating decimal to a fractions?
    (15 votes)
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    • blobby green style avatar for user Joann
      Converting repeating decimals to fractions
      Let x equal the repeating decimal you are trying to convert to a fraction.
      Examine the repeating decimal to find the repeating digit(s).
      Place the repeating digit(s) to the left of the decimal point.
      Place the repeating digit(s) to the right of the decimal point.
      (5 votes)
  • blobby green style avatar for user Dalton Wayne Moore #22
    what does the little line at the top of the number mean
    (7 votes)
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  • hopper cool style avatar for user \«π
    What is 0.78 with the 8 repeating?
    (2 votes)
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    • aqualine ultimate style avatar for user Quatania
      First, you want to create two equations where the repeating digit (in this case the 8) is alone on the right side of the decimal point.

      So for the first equation, multiply both sides by ten, to get:

      10x = 7.888...

      For the second, multiply both sides by 100, to get a different equation with the same repeating eight on the right side of the decimal point:

      100x = 78.888...

      Then subtract the two equations. It helps to see them together:

      100x = 78.888...
      10x = 7.888...

      The repeating 8 is subtracted out, to get:

      90x = 71

      Divide both sides by 90:

      x = 71/90

      71/90 is fully reduced, so that's the answer.

      I hope that helps!
      (4 votes)
  • aqualine ultimate style avatar for user Daniel Hametaj
    Why is math not fun?
    (6 votes)
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  • starky ultimate style avatar for user boopleshmoop
    how would I simplify a repeating decimal like 0.63636363...?
    (5 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Let x = 0.63636363...

      Two digits (63) repeat, so multiply both sides by 10^2=100 to get
      100x = 63.636363...

      Subtracting the first equation from the second equation accomplishes the main goal of canceling out the repeating part:
      99x = 63.

      Dividing both sides by 99 gives
      x = 63/99 = 7/11.

      So 0.63636363... converts to 7/11.
      (5 votes)
  • aqualine ultimate style avatar for user monrael27
    hi how r u guys
    (8 votes)
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  • orange juice squid orange style avatar for user diyanampy
    How would I turn 1.83333... into a simplified fraction?
    (2 votes)
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    • aqualine ultimate style avatar for user Quatania
      The first step is to turn it into two equations with the same decimal.

      So here, to get only the 3 repeating to the right of the decimal point, the first equation would be multiplied by 10, to make:

      10x = 18.333...

      For the second equation, multiplying it by 100 makes sense, to create a different equation. Therefore:

      100x = 183.333...

      Subtract the two equations from each other. Setting it up like this visually makes more sense to me, at least:

      100x = 183.333...
      10x = 18.333...

      The decimal cancels, so it comes out to:

      90x = 165

      Divide both sides by 90 to make it equal to x:

      x = 165/90

      Simplify the fraction by dividing out 15:

      x = 11/6

      I hope that makes sense!
      (9 votes)
  • mr pink orange style avatar for user annabelle
    How would I solve 1.83 with just the three repeating??
    (3 votes)
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  • female robot ada style avatar for user aniza white
    How do we use repeating decimals and is there a better way of explaining it?
    (3 votes)
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    • piceratops ultimate style avatar for user arianasingh
      Repeating decimals are useful for writing numbers like 1/3 as decimals. A way of explaining repeating decimals is that they are similar to pi (π). There is no technical end to the decimal, however unlike with irrational numbers-which we define as real numbers without being a ratio of two numbers- we know the last digit.

      To answer your first question, you need to understand what a decimal really is. A decimal is used to denote rational numbers. It extends the preexisting whole numbers 1,2,3,4,5,6,7,8,9 and 0 (The Hindu-Arabic numerals which are still in use today). Most computers use binary, which only has zeros and ones. Yet, binary is insufficient for division or multiplication. It's also important to realize that software and hardware automatically store numbers like fractions as decimals.

      I hope this answered your question.
      (4 votes)