How to Convert Repeating Decimals to Fractions? (+FREE Worksheet!)

To convert a repeating decimal to a fraction, set the decimal equal to x, multiply by a power of ten that shifts one full repeating block, subtract the original equation to eliminate the repeat, and solve for x. The steps below work through that method on several examples, and a free printable worksheet with answers follows.

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Step by step guide to Convert Repeating Decimals into Fractions

A decimal number with a digit (or group of digits) that repeats forever is a “repeating decimal.” The repeating digits are indicated by drawing a bar over them as in (color{text{ red }}{0.5}) or (color{text{ red }}{0.312}). To convert repeating decimals to fractions:

• Step 1: Let (x) be the repeating decimal.
• Step 2: Find the repeating digit(s) by examining the repeating decimal.
• Step 3: Find (10x) (multiply the repeating decimal by 10) if there is one repeating digit, find (100x) if there are two repeating digits, etc.
• Step 4: Subtract (x) from (10x) (or (100x)) and solve for (x). Note that as you subtract, the difference should be positive for both sides.

Converting Repeating Decimals to Fractions – Example 1:

Convert (color{text{ red }}{0.8888…}) to a fraction.

Solution:

let (x) be the decimal:(x=0.8888…). Since there is one repeating digit, we need to find (10x). Then: (10x=8.888…)

Subtract (x) from (10x), then: (10x-x=8.888…-0.8888…=8)

Solve for (x): (9x=8)→(x=frac{8}{9}). The answer is: (color{text{ red }}{0.8888…=frac{8}{9}})

Converting Repeating Decimals to Fractions – Example 2:

Convert (color{text{ red }}{0.262626…}) to a fraction.

Solution:

let (x) be the decimal:(x=0.2626…). Since there are two repeating digit, we need to find (100x). Then: (100x=26.2626…)

Subtract (x) from (100x), then: (100x-x=26.2626…-0.2626…=26)

Solve for (x): (99x=26)→(x=frac{26}{99}). The answer is: (color{text{ red }}{0.262626…=frac{26}{99}})

Converting Repeating Decimals to Fractions – Example 3:

Convert (color{text{ red }}{0.656565…}) to a fraction.

Solution:

let (x) be the decimal:(x=0.6565…). Since there are two repeating digit, we need to find (100x). Then: (100x=65.6565…)

Subtract (x) from (100x), then: (100x-x=65.6565…-0.6565…=65)

Solve for (x): (99x=65)→(x=frac{65}{99}). The answer is: (color{text{ red }}{0.656565…=frac{65}{99}})

Converting Repeating Decimals to Fractions – Example 4:

Convert (color{text{ red }}{0.393939…}) to a fraction.

Solution:

let (x) be the decimal:(x=0.3939…). Since there are two repeating digit, we need to find (100x). Then: (100x=39.3939…)

Subtract (x) from (100x), then: (100x-x=39.3939…-0.3939…=39)

Solve for (x): (99x=39)→(x=frac{39}{99}). The answer is: (color{text{ red }}{0.393939…=frac{39}{99}})

Exercises for Converting Repeating Decimals to Fractions

Convert each decimal to a fraction.

1. (color{blue}{0.72222…})
2. (color{blue}{0.8888…})
3. (color{blue}{0.767676…})
4. (color{blue}{0.62222…})
5. (color{blue}{0.15555…})
6. (color{blue}{0.37777…})

1. (color{blue}{frac{65}{90}})
2. (color{blue}{frac{8}{9}})
3. (color{blue}{frac{76}{99}})
4. (color{blue}{frac{56}{90}})
5. (color{blue}{frac{14}{90}})
6. (color{blue}{frac{65}{90}})

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