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Converting fractions to recurring decimals

I can divide the numerator of a fraction by its denominator and know that this results in an equivalent recurring decimal.

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New

Converting fractions to recurring decimals

I can divide the numerator of a fraction by its denominator and know that this results in an equivalent recurring decimal.

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Share activities with pupils

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Lesson details

Key learning points

Dividing the numerator by the denominator may result in an equivalent recurring decimal.
It can be shown that 1/9, 1/11 and 1/36 have equivalent recurring decimals.
Using a calculator can help investigate fractions which are equivalent to terminating decimals.
Using a calculator can help investigate fractions which convert to recurring decimals.

Keywords

Recurring decimals - A recurring decimal is one that has an infinite number of digits after the decimal point.

Common misconception

Converting a fraction to a recurring decimal and then rounding the decimal, gives an accurate answer

The use of fractions is more preferred for accuracy than decimals. e.g 1/3 + 1/3 + 1/3 is not 0.3 + 0.3 + 0.3

Get pupils to think if we can multiply a recurring decimal by a number to make it a terminating decimal. Give 1/3, 3/11, 7/9 and 53/154. This thinking task deepens understanding of cancelling prime factors which are not 2 and/or 5.

Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.

Terminating decimals have a number of digits after the decimal point.

Correct Answer: finite

Q2.

Without using a calculator, work out the decimal value of

\frac{13}{20}

Correct Answer: 0.65, 0,65, .65, ,65

Q3.

Select all the fractions that can be written as terminating decimals.

\frac{11}{15}

Correct answer:

\frac{11}{4}

Correct answer:

\frac{11}{8}

Correct answer:

\frac{11}{50}

\frac{11}{30}

Q4.

Aisha uses her calculator to convert

0.668

to a fraction. She gets an answer of

\frac{167}{◻}

. What number should go in the square?

Correct Answer: 250

Q5.

Match each fraction to its terminating decimal. You can use a calculator for this question.

Correct Answer:

\frac{73}{160}

0.45625

$0.45625$

Correct Answer:

\frac{57}{125}

0.456

$0.456$

Correct Answer:

\frac{63}{140}

0.45

$0.45$

Correct Answer:

\frac{2281}{5000}

0.4562

$0.4562$

Correct Answer:

\frac{34}{85}

0.4

$0.4$

Q6.

Sam says that

\frac{231}{330}

is not equivalent to a terminating decimal. Without using a calculator, explain whether Sam is correct or not.

Sam is correct as

330

has factors of

3

and

11

Sam is correct as only fractions with a denominator of

10

terminate

Sam is wrong as

330

is a multiple of

10

Correct answer: Sam is wrong as the fraction simplifies to

\frac{7}{10}

which terminates

Assessment exit quiz

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6 Questions

Q1.

A decimal with an infinite repeating pattern of digits is called a decimal.

Correct Answer: recurring

Q2.

Match each decimal given using dot notation to its equivalent decimal.

Correct Answer:

0. \dot{1} \dot{2}

0.12121212 . . .

$0.12121212 . . .$

Correct Answer:

0. 12 \dot{3}

0.123333333 . . .

$0.123333333 . . .$

Correct Answer:

0. \dot{1} 2 \dot{3}

0.123123123 . . .

$0.123123123 . . .$

Correct Answer:

0. \dot{1}

0.111111111 . . .

$0.111111111 . . .$

Correct Answer:

0. 1 \dot{2} \dot{3}

0.123232323 . . .

$0.123232323 . . .$

Q3.

Select all the fractions which are equivalent to a recurring decimal.

Correct answer:

\frac{1}{3}

Correct answer:

\frac{1}{12}

\frac{3}{24}

\frac{6}{15}

Correct answer:

\frac{10}{12}

Q4.

Use short division to write

\frac{8}{15}

as a decimal.

0.53

0.53333333333

Correct answer:

0.5 \dot{3}

0. \dot{5} \dot{3}

Q5.

Jacob uses his calculator to write

\frac{11}{12}

as a decimal. His calculator display shows the number

0.9166666667

. Jacob says, "This shows my fraction terminates." Is Jacob correct? Explain why.

Yes; Jacob's number has 10 digits after the decimal point so it terminates.

Yes; the last digit is 7 not 6 so it doesn't repeat. So it must terminate.

Correct answer: No; the calculator only has a 12 digit display, the last 6 is rounded up to 7.

No; all decimals recur eventually.

Q6.

Izzy writes the recurring decimal

0.71 \dot{6}

as the fraction

\frac{◻}{60}

. What number should she write in the square?

Correct Answer: 43

Lesson appears in

UnitMaths / Comparing and ordering fractions and decimals (positive and negative)

Maths

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