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

## Step by step guide to Convert Repeating Decimals to 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 \(0.5\) or \(0.312\). 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 \(0.8888…\) 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: \(0.8888…=\frac{8}{9}\)

### Converting Repeating Decimals to Fractions Example 2:

Convert \(0.262626…\) 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: \(0.262626…=\frac{26}{99}\)

### Converting Repeating Decimals to Fractions Example 3:

Convert \(0.656565…\) 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: \(0.656565…=\frac{65}{99}\)

### Converting Repeating Decimals to Fractions Example 4:

Convert \(0.393939…\) 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: \(0.393939…=\frac{39}{99}\)

## 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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