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

This article teaches you how to Convert Repeating Decimals to Fractions in a few simple steps.

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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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