Terminating and recurring decimals

Objective. Recognise terminating and recurring decimal forms of rational numbers.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Terminating and recurring decimals are the two forms a rational number can take when converted from a fraction, depending on the prime factors of the denominator. A decimal terminates if the fraction's denominator, once fully simplified, only has 2 and 5 as prime factors, like ¼ = 0.25. If the denominator has any other prime factor, the decimal recurs forever in a repeating pattern, like ⅓ = 0.333... written as 0.3̇. To decide which type a fraction gives, simplify it fully and check the denominator's prime factors before dividing.

Worked example

Decide whether 5/12 gives a terminating or recurring decimal, explaining using its denominator.

1 $Simplify the fraction: 5/12 is already in lowest terms$
2 $Find the prime factors of the denominator: 12 = 2² \times 3$
3 $Note that 3 is a prime factor other than 2 or 5, so the decimal recurs$
4 $Divide to confirm: 5 \div 12 = 0.41666... = 0.416̇$

Answer: Recurring (0.416̇)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Pupils sometimes assume every fraction with a large denominator must recur, overlooking that 7/8 terminates as 0.875 because 8 = 2³ only contains the prime factor 2.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard