Express The Following Repeating Decimal As A Fraction Calculator

Convert repeating decimals to fractions with our calculator and step-by-step guide. Learn the math behind the conversion and use our formula to solve any repeating decimal.

How to Use This Calculator

This calculator converts repeating decimals to fractions. Simply enter the decimal number and specify the repeating digits, then click "Calculate".

Assumptions

The calculator assumes the repeating pattern starts immediately after the decimal point. For example, 0.333... is 1/3, not 0.1333... which would be 1/75.

The Conversion Method

To convert a repeating decimal to a fraction, follow these steps:

Let x = the repeating decimal
Multiply x by 10^n where n is the number of repeating digits
Subtract the original x from this new equation
Solve for x

Formula

For a repeating decimal 0.a̅b̅c̅... (where abc... repeats):

x = 0.a̅b̅c̅...

10^n x = abc̅.abc̅...

(10^n - 1)x = abc̅ - 0.a̅b̅c̅...

x = abc̅ / (10^n - 1)

Worked Examples

Example 1: 0.333...

Let x = 0.333...

10x = 3.333...

Subtract: 10x - x = 3.333... - 0.333...

9x = 3

x = 3/9 = 1/3

Example 2: 0.142857...

Let x = 0.142857...

1000000x = 142857.142857...

Subtract: 1000000x - x = 142857.142857... - 0.142857...

999999x = 142857

x = 142857/999999 = 1/7

Frequently Asked Questions

What if the repeating decimal has non-repeating digits?: For decimals like 0.1666..., first convert the non-repeating part (0.1) to a fraction (1/10), then convert the repeating part (0.0666...) separately and add the results.
Can this calculator handle mixed repeating decimals?: Yes, but you'll need to separate the repeating and non-repeating parts. For example, 0.12345̅6̅ would require converting 0.123 and 0.00045̅6̅ separately.
What if the repeating decimal doesn't start right after the decimal point?: You'll need to adjust the formula accordingly. For example, 0.1333... would require multiplying by 1000 instead of 10 to align the repeating parts.