# How to Convert Repeating Decimals to Fractions

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.

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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