M8MATH-1.1

Repeating Decimals to Fractions

Learn to convert repeating decimals to fractions using the algebraic subtraction method. Master one-digit, two-digit, and mixed repeating patterns.

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What this lesson covers

You already know how to convert a simple fraction like 34\frac{3}{4} into a decimal: just divide 3 by 4 to get 0.75. But what about working backward? If you see a decimal like 0.36‾0.\overline{36} (where 36 repeats forever), how do you turn it back into a fraction? This lesson teaches you a clever algebraic trick that works every time, turning infinite repeating decimals into exact fractions. Understanding this connection is the key to seeing why repeating decimals are actually rational numbers.

What Makes a Decimal Repeating or Terminating

Every rational number (a number that can be written as a fraction) produces a decimal that either terminates or repeats. A terminating decimal stops, like 14=0.25\frac{1}{4} = 0.25. A repeating decimal goes on forever with a pattern, like 13=0.333...\frac{1}{3} = 0.333... or 211=0.181818...\frac{2}{11} = 0.181818.... We use a bar over the digits to show which digits repeat: 0.3‾0.\overline{3} means 0.3333…, and 0.18‾0.\overline{18} means 0.181818…. The repeating block can be one digit, two digits, or more. Some decimals have a non-repeating part at the start before the pattern kicks in, like 0.83‾=0.8333...0.8\overline{3} = 0.8333..., where the 8 appears once and then 3 repeats. Understanding which decimal pattern you're looking at is the first step to converting it correctly.

The Subtract-and-Divide Method: One-Digit Repeats

The algebraic method for converting repeating decimals uses multiplication and subtraction to eliminate the infinite pattern. Let's convert 0.7‾0.\overline{7} to a fraction. Set x=0.777...x = 0.777.... Since one digit repeats, multiply both sides by 10: 10x=7.777...10x = 7.777.... Now subtract the original equation from this new one:10x−x=7.777...−0.777...10x - x = 7.777... - 0.777...9x=79x = 7x=79x = \frac{7}{9}Why does this work? When you subtract, the infinite repeating tails cancel out, leaving you with a whole number. The key is choosing the right power of 10: multiply by 10n10^n where nn is the number of digits in the repeating block. For a one-digit repeat, multiply by 10. For a two-digit repeat, multiply by 100. The denominator you get is always 9's: one digit repeats gives denominator 9, two digits gives denominator 99, three digits gives denominator 999.

The Subtract-and-Divide Method: Two-Digit Repeats

When two digits repeat, the process is the same but you multiply by 100 instead of 10. Convert 0.36‾0.\overline{36} to a fraction. Set x=0.363636...x = 0.363636.... Multiply by 100100 (since 2 digits repeat): 100x=36.363636...100x = 36.363636.... Subtract:100x−x=36.363636...−0.363636...100x - x = 36.363636... - 0.363636...99x=3699x = 36x=3699x = \frac{36}{99}Simplify by finding the GCD of 36 and 99, which is 9:x=36÷999÷9=411x = \frac{36 \div 9}{99 \div 9} = \frac{4}{11}Always simplify your final fraction. The denominator 99 comes from having two repeating digits: it's 102−1=100−1=9910^2 - 1 = 100 - 1 = 99. More generally, nn repeating digits give a denominator of 10n−110^n - 1 (which is a string of nn nines). Understanding this pattern makes it easier to remember why the method works.

Mixed Repeating Decimals: Non-Repeating Lead Digits

Some decimals have digits that don't repeat at the start. For example, 0.83‾=0.8333...0.8\overline{3} = 0.8333... has the 8 before the 3's begin repeating. The method adapts slightly: you need two equations to handle both the non-repeating and repeating parts. Set x=0.8333...x = 0.8333.... The non-repeating part has 1 digit, so multiply by 10: 10x=8.333...10x = 8.333.... The repeating part has 1 digit, so multiply by an additional factor of 10: 100x=83.333...100x = 83.333.... Now subtract the first equation from the second:100x−10x=83.333...−8.333...100x - 10x = 83.333... - 8.333...90x=7590x = 75x=7590=56x = \frac{75}{90} = \frac{5}{6}The denominator 90 is 102−101=100−1010^2 - 10^1 = 100 - 10. In general, if mm digits don't repeat and nn digits do repeat, the denominator is 10m+n−10m10^{m+n} - 10^m. This formula is a string of nn nines followed by mm zeros. The key insight: always multiply by the highest power of 10 you need to align the repeating blocks, then subtract to cancel them.

Checking Your Work and Simplifying

After you convert a repeating decimal to a fraction, always check by dividing the numerator by the denominator to see if you get the original decimal back. For instance, if you got 56\frac{5}{6}, divide: 5÷6=0.8333...5 \div 6 = 0.8333..., which matches 0.83‾0.8\overline{3}. Also, always simplify your fraction to lowest terms before you call it done. Finding the GCD of numerator and denominator ensures your final answer is in the clearest form. If you're not sure whether you've simplified fully, factor both numbers and cancel all common factors. This habit prevents careless errors and makes your answer easier to verify.

Key terms

Repeating decimal.
A decimal that has a block of digits that repeats infinitely, written with a bar over the repeating block, such as 0.27‾0.\overline{27} for 0.272727….
Terminating decimal.
A decimal that ends after a finite number of digits, such as 0.25 or 0.5, which can always be written as a fraction with a denominator that is a power of 10.
Rational number.
A number that can be expressed as a fraction ab\frac{a}{b} where aa and bb are integers and b≠0b \neq 0; its decimal form either terminates or repeats.
Non-repeating lead digit.
A digit or group of digits at the beginning of a decimal that appear once before the repeating block starts, as in 0.83‾0.8\overline{3}.
Bar notation.
The symbol placed over one or more digits in a decimal to indicate that those digits repeat infinitely; for example, the bar over 36 in 0.36‾0.\overline{36} shows that 36 repeats.
Subtract-and-divide method.
An algebraic technique that converts a repeating decimal to a fraction by setting up an equation, multiplying by an appropriate power of 10, subtracting to eliminate the infinite tail, and then solving for the variable.

Worked example

Convert 0.523‾0.5\overline{23} to a fraction in simplest form.
This decimal has a non-repeating lead digit (5) followed by a repeating block (23). We need two equations to handle both parts.

Step 1: Let x=0.5232323...x = 0.5232323...

Step 2: Count the non-repeating digits: 1 (just the 5). Multiply by 101=1010^1 = 10:10x=5.232323...10x = 5.232323...Step 3: Count the total non-repeating plus repeating: 1 + 2 = 3 digits. Multiply by 103=100010^3 = 1000:1000x=523.232323...1000x = 523.232323...Step 4: Subtract the first equation from the third to cancel the repeating tail:1000x−10x=523.232323...−5.232323...1000x - 10x = 523.232323... - 5.232323...990x=518990x = 518Step 5: Solve for xx:x=518990x = \frac{518}{990}Step 6: Simplify by finding the GCD of 518 and 990. Both are even, so divide by 2:x=259495x = \frac{259}{495}Check if there are more common factors. 259 = 7 × 37 and 495 = 5 × 9 × 11 = 5 × 99, so they share no more factors.x=259495x = \frac{259}{495}Step 7: Verify by dividing: 259÷495=0.5232323...259 \div 495 = 0.5232323..., which matches the original. ✓

Practice questions

Convert 0.45‾0.\overline{45} to a fraction in simplest form.
  1. 4599\frac{45}{99}
  2. 511\frac{5}{11}
  3. 920\frac{9}{20}
  4. 45100\frac{45}{100}

Answer: 511\frac{5}{11}

Set x=0.454545...x = 0.454545.... Since two digits repeat, multiply by 100: 100x=45.454545...100x = 45.454545.... Subtract: 100x−x=45100x - x = 45, so 99x=4599x = 45, giving x=4599x = \frac{45}{99}. Now simplify: the GCD of 45 and 99 is 9. Divide both by 9: 45÷999÷9=511\frac{45 \div 9}{99 \div 9} = \frac{5}{11}. You can verify by dividing 5 by 11, which gives 0.454545…. The other choices represent common mistakes: forgetting to simplify (4599\frac{45}{99}), confusing it with a terminating decimal (920\frac{9}{20}), or treating it as a non-repeating decimal (45100\frac{45}{100}).
What is 0.6‾0.\overline{6} as a fraction? Show your work.

Answer: 23\frac{2}{3}

Set x=0.666...x = 0.666.... Since one digit repeats, multiply by 10: 10x=6.666...10x = 6.666.... Subtract the original equation: 10x−x=6.666...−0.666...10x - x = 6.666... - 0.666..., so 9x=69x = 6. Therefore, x=69x = \frac{6}{9}. Simplify by dividing both numerator and denominator by their GCD, which is 3: x=23x = \frac{2}{3}. Students sometimes forget to simplify and leave the answer as 69\frac{6}{9}, which is not in lowest terms. You can check by dividing 2 by 3 on a calculator to confirm you get 0.666….

FAQ

Why do we multiply by a power of 10 when converting repeating decimals?
Multiplying by a power of 10 shifts the decimal point so that the repeating blocks line up. When you subtract two equations with aligned repeating blocks, the infinite tails cancel out, leaving you with a whole number on the right side. This turns an infinite problem into a finite one you can actually solve.
What is the denominator always going to be?
The denominator is always a string of 9's and 0's. For a pure repeating decimal with nn repeating digits, the denominator is nn nines (for example, one digit repeating gives 9, two digits give 99). For a decimal with non-repeating lead digits, the denominator is nn nines followed by mm zeros, where nn is the number of repeating digits and mm is the number of non-repeating digits.
Do I need to simplify my fraction at the end?
Yes, always. Your final answer should be a fraction in lowest terms, meaning the numerator and denominator share no common factors other than 1. To simplify, find the GCD and divide both the top and bottom by it. An unsimplified fraction is not a complete answer.
How can I check if my conversion is correct?
Divide the numerator by the denominator using long division or a calculator. You should get back the original repeating decimal. For example, if you converted 0.36‾0.\overline{36} to 411\frac{4}{11}, compute 4÷114 \div 11 and verify it equals 0.363636….

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