Fractions and Recurring Decimals Revision Notes for Edexcel GCSE Maths

Fractions and recurring decimals

Understanding recurring and terminating decimals

When you convert a fraction to a decimal, you'll get one of two types: a terminating decimal or a recurring decimal. Understanding which type you'll get depends on the prime factors in the denominator of your fraction.

Recurring decimals have a pattern of numbers that repeats forever. For example, $\frac{1}{3}$ becomes $0.333333...$ The repeating part is usually shown with dots or a bar above the repeating digits.

infoNote

It's important to note that it doesn't have to be just one digit that repeats - you could have patterns like $0.142857142857...$ where the whole sequence "142857" repeats.

Terminating decimals are finite - they come to an end. For example, $\frac{1}{2}$ becomes $0.5$ , which stops there.

The key to predicting which type you'll get lies in examining the prime factors of the denominator when your fraction is in its simplest form.

chatImportant

If the denominator only contains the prime factors 2 and 5, you'll get a terminating decimal. If the denominator contains any other prime factors, you'll get a recurring decimal.

This relationship exists because our decimal system is based on powers of 10, and $10 = 2 \times 5$ . Only fractions whose denominators divide evenly into powers of 10 will terminate.

Converting recurring decimals to fractions

Converting recurring decimals back to fractions requires a clever algebraic approach. There are two main scenarios you'll encounter.

Basic method (recurring starts immediately)

When the recurring pattern starts immediately after the decimal point, use this method:

lightbulbExample

Worked Example: Converting 0.234̄ to a fraction

Step 1: Let $r = 0.234234234...$

Step 2: Multiply by 1000 (to move past the three-digit repetition):

$1000r = 234.234234234...$

Step 3: Subtract the original equation:

$1000r - r = 234.234234... - 0.234234...$

$999r = 234$

Step 4: Solve for r:

$r = \frac{234}{999} = \frac{26}{111}$

The process involves:

Name your decimal (let's call it r)
Multiply r by a power of 10 to move the decimal point past one complete repetition
Subtract the original equation to eliminate the decimal part
Solve for r and simplify if possible

Trickier method (non-repeating part first)

When there's a non-repeating part before the recurring section begins, you need to account for this:

The process involves:

Name your decimal
Multiply by a power of 10 to move past the non-repeating part
Multiply again to move past one complete repetition of the recurring part
Subtract to eliminate the decimal part
Solve for r and simplify

This method handles decimals like $0.1\overline{6}$ where the 1 doesn't repeat but the 6 does.

Quick method for exams

For converting recurring decimals to fractions quickly, you can use the "Just Learning the Result" method:

infoNote

The recurring part goes on top
The same number of 9s goes on the bottom
But this only works if the recurring part starts immediately after the decimal point

Converting fractions to recurring decimals

You might encounter questions asking you to convert fractions to recurring decimals. There are two approaches:

Method 1: Equivalent fraction approach

Find an equivalent fraction with all 9s in the denominator. The numerator will tell you the recurring part.

lightbulbExample

Worked Example: Converting 8/33 to a recurring decimal

Step 1: Find equivalent fraction with 9s in denominator:

$\frac{8}{33} = \frac{24}{99}$

Step 2: The numerator shows the recurring part:

$\frac{24}{99} = 0.\overline{24}$

Method 2: Long division

Simply divide the numerator by the denominator using long division. The pattern will eventually repeat, giving you the recurring decimal directly.

The first method is often quicker for simple fractions, while long division is more reliable for complex cases or when using a calculator.

bookmarkSummary

Key Points to Remember:

Fractions with denominators containing only prime factors 2 and 5 give terminating decimals
All other fractions produce recurring decimals
To convert recurring decimals to fractions, multiply by appropriate powers of 10 and subtract to eliminate the decimal part
The "number of 9s" method works quickly for basic recurring decimals that start immediately
You can convert fractions to recurring decimals using equivalent fractions with 9s in the denominator or by long division

Fractions and Recurring Decimals (Edexcel GCSE Maths): Revision Notes