Recurring Decimals – Part 2

Read Recurring Decimals – Part 1

Once students can convert fractions to recurring decimals by division and convert straightforward recurring decimals to fractions, for example,

fig 1

they should try the more complex recurring decimals, such as,

fig 2

Exam papers tend to either ask these questions in a straightforward manner,

fig 3


Or a common approach is a two part question,

fig 4



Part (a) forces students to convert
fig 5

The hence in (b) suggests writing
fig 6

Obviously this method can be used even if the structure is not given but many students write
fig 8

they then often find the subtraction difficult as they are subtracting infinite decimals.

It can be easier to see this if it is written out as a column subtraction

fig 9

Hopefully students will see that the 4s go when subtracted

fig 10

This leaves

fig 11

As this is not a conventional fraction (as it contains a decimal) we have to find an equivalent fraction that does not include decimals. The obvious method is to multiply numerator and denominator by 10 (or 100, 1000 etc. if there is more than one decimal place.)

fig 12

My students enjoyed calling this ‘cancelling up’!

You will need to make a call as to whether to teach both methods and allow your students to choose which they prefer or just to teach one. This will probably depend on the ability of your class.

Many thanks,

Steve Cavill

Steve Cavill BSc(Steve CavillHons) PGCE FCIEA has taught maths in both state and independent schools. He spent a few years as an Associate Lecturer for the OU and has written a number of GCSE maths books, workbooks and revision guides as well as being a senior examiner and moderator for GCSE and IGCSE.