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

Fractions and recurring decimals

What are recurring and terminating decimals?

When you convert a fraction to a decimal, you'll get one of two types of results. Understanding the difference between these is crucial for your GCSE maths success.

Recurring decimals are decimals where a pattern of digits repeats endlessly. For example, $\frac{1}{3} = 0.333333...$ where the 3 keeps repeating forever. We show this repetition using dot notation - so $\frac{1}{3} = 0.\overline{3}$ where the dot above the 3 shows it repeats.

Terminating decimals are decimals that come to an end. For example, $\frac{1}{4} = 0.25$ - there are no more digits after the 5, so it terminates.

The prime factor rule

Here's the key rule that determines whether a fraction will give you a terminating or recurring decimal:

Let's see this rule in action:

This table clearly shows the pattern. Fractions like $\frac{1}{5}$, $\frac{1}{2}$, and $\frac{1}{20}$ all have denominators that only contain the prime factors 2 and 5, so they give terminating decimals. Meanwhile, fractions like $\frac{1}{3}$, $\frac{1}{7}$, and $\frac{1}{6}$ have denominators with other prime factors (3, 7, and 3 respectively), so they give recurring decimals.

Converting recurring decimals to fractions

Converting a recurring decimal back to a fraction might seem tricky at first, but there's a systematic method that works every time. The approach depends on whether the recurring part starts immediately after the decimal point or not.

Basic method (immediate recurrence)

When the recurring part starts straight after the decimal point, like $0.\overline{234}$, the algebraic method provides a reliable solution:

The method works by using algebra to eliminate the decimal part through strategic multiplication and subtraction.

Trickier method (delayed recurrence)

When there's a non-repeating part before the recurring section, like $0.1\overline{6}$, you need an extra step:

Converting fractions to recurring decimals

Sometimes you'll need to go the other way - from fractions to recurring decimals. There are two main approaches:

Method 1: Find an equivalent fraction with all 9s on the bottom If you can make the denominator consist entirely of 9s, the numerator will tell you the recurring part directly. For example, $\frac{8}{33}$ can be rewritten as $\frac{24}{99}$, which equals $0.\overline{24}$.

Method 2: Use division You can simply divide the numerator by the denominator. If you're allowed a calculator, this is straightforward. If not, you can use long division, though this can be time-consuming.