How to Convert Recurring Decimals

Recurring decimals are decimals that repeat a pattern of digits indefinitely. Understanding how to convert these decimals into fractions is an essential skill for your GCSE Maths exams. This guide will explain the key points about recurring decimals, how to use dot notation, and the steps to convert recurring decimals into fractions.

Key Points about Recurring Decimals

• A recurring decimal is a decimal that repeats the same block of digits infinitely. For example, $0.\overline{3}$ (0.3333...) is a recurring decimal because the digit "3" repeats infinitely.
• Dot notation or bar notation is used to represent recurring decimals. A recurring decimal can be written in two ways:
  • Dot notation: Where a dot is placed over the repeating part of the decimal, e.g., $0.\overline{3}$ represents 0.3333...
  • Bar notation: Where a horizontal bar is placed over the repeating digits, e.g., $0.\overline{3}$ means the digit "3" repeats indefinitely.
• The repeating part is known as the recurring part or repeating block. For example, in $0.\overline{142857}$, the recurring part is "142857."

How to Use Dot Notation for Recurring Decimals

Dot notation is a simple way to represent recurring decimals. The repeating block is indicated by a dot placed above the first repeating digit.

• Example: $0.\overline{6}$ means the digit "6" repeats infinitely (0.6666...).
• Example: $0.\overline{142857}$ means the block "142857" repeats indefinitely.

Steps to Convert Recurring Decimals into Fractions

There are specific steps to convert recurring decimals to fractions. Below are the methods you can use for both recurring decimals with a single repeating digit and recurring decimals with multiple repeating digits.

Converting a Recurring Decimal with a Single Repeating Digit

Consider the recurring decimal $x = 0.\overline{3}$ (where "3" repeats infinitely).

Steps:

1. Let $x = 0.\overline{3}$

2. Multiply both sides of the equation by 10 (this moves the decimal point one place to the right):

   $$10x = 3.\overline{3}$$

3. Subtract the original equation ($x = 0.\overline{3}$ from the new equation ($10x = 3.\overline{3}$ to eliminate the repeating part:

   $$10x - x = 3.\overline{3} - 0.\overline{3}$$

   $$9x=3$$

4. Solve for $x$:

$$x = \frac{3}{9} = \frac{1}{3}$$​

Thus, $0.\overline{3} = \frac{1}{3}$​.

Converting a Recurring Decimal with Multiple Repeating Digits

Consider the recurring decimal $x = 0.\overline{1428}$ (where the block "1428" repeats infinitely).

Steps:

1. Let $x = 0.\overline{1428}$.

2. Multiply both sides by 1,000,000 (because there are six repeating digits):

   $$1000000x = 1428.\overline{1428}$$

3. Subtract the original equation ($x = 0.\overline{1428}$) from the new equation ($1000000x = 1428.\overline{1428}$):

   $$1000000x - x = 1428.\overline{1428} - 0.\overline{1428}$$

   $$ 999999x=1428$$

4. Solve for $x$:

   $$x = \frac{1428}{999999}$$

5. Simplify the fraction, if possible:

   $$x = \frac{1}{7}$$

Thus, $0.\overline{1428} = \frac{1}{7}$​.

Examples of Recurring Decimal Conversions

Example 1: Convert $0.\overline{1}$ to a fraction.

1. Let $x = 0.\overline{1}$.
2. Multiply by 10: $10x = 1.\overline{1}$

3. Subtract the original equation from the new equation: 

   $$10x - x = 1.\overline{1} - 0.\overline{1}$$

   $$ 9x=1$$

4. Solve for $x$: 

   $$x = \frac{1}{9}$$

So, $0.\overline{1} = \frac{1}{9}$​.

Example 2: Convert $0.\overline{27}$ to a fraction.

1. Let $x = 0.\overline{27}$.

2. Multiply by 100 (since there are two repeating digits): 

   $$100x = 27.\overline{27}$$

3. Subtract the original equation from the new equation: 

   $$100x - x = 27.\overline{27} - 0.\overline{27}$$

   $$99x = 27$$

4. Solve for $x$: 

$$x = \frac{27}{99} = \frac{3}{11}$$

So, $0.\overline{27} = \frac{3}{11}$​.

Summary of Steps to Convert Recurring Decimals

By mastering these techniques, you'll be well-prepared to handle recurring decimals in your GCSE Maths exams.