Topic Eight (1.8): Terminating and Recurring Decimals

 Introduction

There are two types of decimals:  

Terminating Decimal: a decimal that ends (e.g., 0.25 or 0.5); and  

Recurring Decimal: a decimal that repeats in a pattern forever (e.g., 0.333… or 0.142857142857…).  

The video below explains these two types: 

In this lesson we will look at how to recognise whether a fraction is equivalent to a recurring or a terminating decimal.  

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Warm Up

Decimal Detective  

For each fraction below: 

1. Predict whether the decimal equivalent will terminate or recur. 
2. Use a calculator to convert each fraction into a decimal and copy and fill in the table below. Compare the results with your predictions.

Fraction	Predict: Terminate or Recur?	Decimal Equivalent	Terminate or Recur?	Correct Prediction?
$\frac{5}{2}$
$\frac{17}{5}$
$\frac{8}{15}$
$\frac{9}{20}$
$\frac{17}{6}$
$\frac{11}{21}$

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Working through your course

Watch this excellent video, which discusses the Warm up activity and generalises the results 

Summary 

Step 1: Simplify the Fraction 

Always simplify the fraction to its lowest terms. For example, simplify $\frac{6}{15}$ to $\frac{2}{5}$.

Step 2: Examine the Denominator 

The behaviour of the decimal depends on the denominator (after simplification). 

• If the denominator has only the prime factors 2 and/or 5, the decimal will terminate. 

• If the denominator has any other prime factors, the decimal will recur. 

Examples: 

1. Terminating Decimals: 

• • • $\frac{1}{2}$= 0.5; Denominator = 2.

• • • $\frac{3}{8}$= 0.375; Denominator = 2³.

    • $\frac{7}{10}$ = 0.7; Denominator = 2 × 5.

  2.  Recurring Decimals: 

• • • $\frac{1}{3}$= 0.33333….; Denominator = 3 (not 2 or 5).

    • $\frac{2}{7}$= 0.285714285714285714…; Denominator = 7.

    • $\frac{5}{11}$= 0.4545454….; Denominator = 11.

 

Copy and complete this table below, which is similar to the warm-up activity, but choosing your own fractions (you can make these as tricky as you wish!).  Use the method described above to predict whether the decimals will terminate or recur, and use a calculator to check your predictions. 

Fraction	Predict: Terminate or Recur?	Decimal Equivalent	Terminate or Recur?	Correct Prediction?

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Plenary

This game will help you practise prime factorisation: 

PLAY - Prime factorisation - pairs activity (MyMaths) Links to an external site. [Not available on sample course]

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Support activity for this topic

If you need a reminder about how to write a number as a product of its prime factors revisit Topic 1.3. 

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Extension activity for this topic

Can you explain why decimals terminate if the denominator has only the prime factors 2 and/or 5?