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.
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 \(\color{red}{0.5}\) or \(\color{red}{0.312}\). 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 \(\color{red}{0.8888…}\) 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: \(\color{red}{0.8888…=\frac{8}{9}}\)
Converting Repeating Decimals to Fractions – Example 2:
Convert \(\color{red}{0.262626…}\) 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: \(\color{red}{0.262626…=\frac{26}{99}}\)
Converting Repeating Decimals to Fractions – Example 3:
Convert \(\color{red}{0.656565…}\) 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: \(\color{red}{0.656565…=\frac{65}{99}}\)
Converting Repeating Decimals to Fractions – Example 4:
Convert \(\color{red}{0.393939…}\) 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: \(\color{red}{0.393939…=\frac{39}{99}}\)
Exercises for Converting Repeating Decimals to Fractions
Convert each decimal to a fraction.
- \(\color{blue}{0.72222…}\)
- \(\color{blue}{0.8888…}\)
- \(\color{blue}{0.767676…}\)
- \(\color{blue}{0.62222…}\)
- \(\color{blue}{0.15555…}\)
- \(\color{blue}{0.37777…}\)
- \(\color{blue}{\frac{65}{90}}\)
- \(\color{blue}{\frac{8}{9}}\)
- \(\color{blue}{\frac{76}{99}}\)
- \(\color{blue}{\frac{56}{90}}\)
- \(\color{blue}{\frac{14}{90}}\)
- \(\color{blue}{\frac{65}{90}}\)
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