Repeating Decimal to Fraction Calculator

How to Convert Repeating Decimal to Fraction

To convert a repeating decimal into a fraction, we follow a structured algebraic approach that isolates the repeating part and eliminates it through subtraction. This method works for both pure repeating decimals (like 0.3̅) and mixed repeating decimals (like 1.2̅3̅).

The general steps involve:

• Representing the repeating decimal as a variable.

• Multiplying it by a power of 10 to shift the decimal point to align repeating digits.

• Creating and subtracting two equations to eliminate the repeating part.

• Solving for the variable and simplifying the result into a proper or mixed fraction.

Let’s look at a few different types of repeating decimals and how they’re converted:

Example 1: Convert Mixed Repeating Decimal to Fraction

Decimal: 2.45

Step-by-step:

1. Original repeating decimal: 2.457

2. Let X = 2.457

3. Multiply by 10 to move repeating digit after decimal:

  10X = 24.577

4. Subtract equation (1) from equation (2)

  10X – 1X = 24.57 – 2.457
               9X = 22.12

6. Solve for X:

  X = 22.12⁄9

Simplify & convert to mixed fraction

Result:

2.457 = 2212⁄900 = 2 103⁄225