0. 0 9 is the recurring decimal 0.090909..., in which the block 09 repeats. Write it as a fraction in its simplest form. Give the numerator and the denominator.

numerator = 1, denominator = 11

0.5 3 is the recurring decimal 0.5333..., in which only the 3 repeats. Write it as a fraction in its simplest form. Give the numerator and the denominator.

numerator = 8, denominator = 15

Work out 0. 4 multiplied by 0. 5, where 0. 4 is 0.4444... with the 4 repeating and 0. 5 is 0.5555... with the 5 repeating. Give your answer as a fraction in its simplest form, stating the numerator and the denominator.

numerator = 20, denominator = 81

0.2 3 6 is the recurring decimal 0.2363636..., in which the block 36 repeats but the 2 does not. Write it as a fraction in its simplest form. Give the numerator and the denominator.

numerator = 13, denominator = 55

Are recurring decimals on the Foundation paper?

Not the conversion. Spec ref N10 is split: working with terminating decimals and their fractions is on both tiers, while changing a recurring decimal into a fraction and back is marked as Higher tier content in 1MA1. A Foundation paper can still show a decimal in dot notation or ask which fractions recur, but it will not ask for the algebraic conversion.

How many marks is a recurring decimal question worth on Edexcel Higher?

Usually 2-3 marks. A single repeating block is normally 2-3, and a decimal with one or two non-recurring digits in front of the block is typically 3-4 because it needs two multiplications. It tends to appear once per series rather than once per paper, so treat it as a topic to bank rather than one to rely on.

Can I just divide on the calculator instead of using algebra?

Only when the question does not ask you to prove anything. Edexcel almost always writes "prove algebraically", "use algebra to show" or "show clear algebraic working" on these, and then the marks are for the x line, the multiplied line and the subtraction. A calculator display cannot show that the recurring tail cancels.

Is 0.9 recurring, the decimal 0.9999... with the 9 repeating forever, really equal to 1?

Yes, and the standard method proves it. Let x be 0.9999..., then 10x is 9.9999..., and subtracting gives 9x = 9, so x = 1. The two are different names for the same number rather than two numbers that are very close together.

How do I tell whether a fraction gives a terminating or a recurring decimal?

Cancel the fraction fully, then factorise the denominator. If the only prime factors are 2 and 5 the decimal terminates, because the denominator divides a power of 10. Any other prime factor, such as the 3 in 12 or the 7 in 14, makes the decimal recur.

Recurring Decimals - Edexcel GCSE Higher Maths

Typically 2-4 marks when it appears, usually a single question in the second half of a paper · Spec 1MA1

A recurring decimal is one whose digits repeat forever in a fixed block. Dots mark the first and last digit of that block, so $0.\dot{4}$ is 0.4444... with the 4 repeating, and $0.\dot{1}\dot{8}$ is 0.181818... with the block 18 repeating. Edexcel files this under spec ref N10, which also covers terminating decimals and their fractions. The terminating half sits on both tiers; changing a recurring decimal into a fraction and back again is Higher tier only, so a Foundation candidate may have to read the dot notation but will not be asked to convert.

Edexcel usually sets it as a short standalone question worth 2-3 marks, somewhere in the second half of the paper, and the wording is the part to read carefully. "Prove algebraically", "Use algebra to show" and "You must show clear algebraic working" all amount to the same instruction: a correct fraction with nothing above it scores zero. Recurring decimals can turn up on any of the three papers. Paper 1 is non-calculator, and a calculator does surprisingly little on Papers 2 and 3, because a display reading 0.4242424 is not working and the cancelling at the end takes one line by hand.

Two slips account for most of the lost marks. The first is the power of ten: a two-digit repeating block needs $100x$ rather than $10x$ , and a decimal such as $0.2\dot{7}$ , where one digit sits before the block, needs both $10x$ and $100x$ so that the recurring tails line up before you subtract. The second is simplest form. Writing $\frac{42}{99}$ earns the method marks and then loses the accuracy mark on a question that asked for lowest terms; 99 factorises as 9 times 11, so test 3, 9 and 11 before you stop. Practise writing all three lines, the $x$ line, the multiplied line and the subtraction, even on a decimal whose fraction you already know.

Worked example

Prove algebraically that the recurring decimal

0.3\dot{1}\dot{8}

, in which the block 18 repeats so that the decimal is 0.3181818..., can be written as

\frac{7}{22}

[4 marks]

Let $x = 0.3181818\ldots$ , writing out enough digits to show the pattern.
M1: naming the decimal as $x$ and writing the repeat out; this line alone is worth a mark on Edexcel schemes
One digit (the 3) comes before the block, and the block 18 is two digits long. Multiply by 10 and by 1000 so that both lines have the same recurring tail: $10x = 3.181818\ldots$ $1000x = 318.181818\ldots$
M1: two correct multiples whose decimal parts match; $10x$ shifts past the non-recurring 3, then $\times 100$ for the two-digit block
Subtract the smaller from the larger. The recurring tails cancel exactly: $1000x - 10x = 318.181818\ldots - 3.181818\ldots$ $990x = 315$
M1: a correct subtraction leaving whole numbers on both sides
Divide by 990, then cancel by 45: $x = \frac{315}{990} = \frac{7}{22}$ , which is what was to be shown.
A1: fully cancelled and linked back to the given fraction

Practice questions

These are original questions in Edexcel 1MA1 Higher style, covering the conversion in both directions and the mixed decimals where only part of the number recurs. Every recurring decimal is given twice, in dot notation and in words, so there is never any doubt about which digits repeat. Where a question asks for a fraction in its simplest form, type the numerator and the denominator into the two labelled boxes; work them without a calculator, as you would on Paper 1.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [2 marks]

One of these fractions can be written as a terminating decimal. Which one?

Answer options for question 1
$\frac{7}{12}$ $\frac{3}{16}$ $\frac{5}{9}$ $\frac{4}{15}$
Question 2 Warm-up [2 marks]

$0.\dot{0}\dot{9}$ is the recurring decimal 0.090909..., in which the block 09 repeats. Write it as a fraction in its simplest form. Give the numerator and the denominator.

numerator = denominator =
Question 3 Exam pace [3 marks]

$0.5\dot{3}$ is the recurring decimal 0.5333..., in which only the 3 repeats. Write it as a fraction in its simplest form. Give the numerator and the denominator.

numerator = denominator =
Question 4 Exam pace [3 marks]

One of these fractions can only be written as a recurring decimal. Which one?

Answer options for question 4
$\frac{9}{40}$ $\frac{7}{24}$ $\frac{13}{50}$ $\frac{21}{60}$
Question 5 Stretch [3 marks]

Work out $0.\dot{4}$ multiplied by $0.\dot{5}$ , where $0.\dot{4}$ is 0.4444... with the 4 repeating and $0.\dot{5}$ is 0.5555... with the 5 repeating. Give your answer as a fraction in its simplest form, stating the numerator and the denominator.

numerator = denominator =
Question 6 Stretch [4 marks]

$0.2\dot{3}\dot{6}$ is the recurring decimal 0.2363636..., in which the block 36 repeats but the 2 does not. Write it as a fraction in its simplest form. Give the numerator and the denominator.

numerator = denominator =

Common mistakes examiners see

Multiplying by 10 when the repeating block has two digits, so $0.\dot{1}\dot{3}$ gives $10x = 1.313131\ldots$ and the subtraction $10x - x$ leaves $9x = 1.181818\ldots$ , another recurring decimal that cannot be simplified cleanly.

Count the digits in the block first and match the power of ten to it: one digit means $10x$ , two digits mean $100x$ , three digits mean $1000x$ . The test is whether the two lines have identical digits after the decimal point before you subtract.
Stopping at the unsimplified fraction, handing in $\frac{42}{99}$ or $\frac{18}{99}$ when the question said "in its simplest form".

Cancel before you write the final line. The denominators that come out of this method are 9, 99, 999, 90 and 990, and their factors are small, so check 3, 9 and 11 in turn. $\frac{42}{99}$ becomes $\frac{14}{33}$ and $\frac{18}{99}$ becomes $\frac{2}{11}$ .
On a decimal with a non-recurring digit in front, such as $0.2\dot{8}$ , subtracting $x$ from $10x$ to get $9x = 2.6$ and then writing $x = \frac{2.6}{9}$ .

A fraction with a decimal in it is not an answer. Either multiply top and bottom by 10 to get $\frac{26}{90} = \frac{13}{45}$ , or avoid the problem by using $100x - 10x = 26$ from the start, which keeps both sides whole.
Treating the decimal as if it stopped, so $0.\dot{4}\dot{2}$ is converted as $\frac{42}{100} = \frac{21}{50}$ .

Write the decimal out to six or seven digits before you start. A recurring decimal is an exact value, not a rounded one, and the whole method depends on the tail being infinite so that it cancels in the subtraction.
Writing down the right fraction straight from the "digits over nines" shortcut on a question that says "prove algebraically" or "show clear algebraic working".

Set out the three lines: $x = \ldots$ , the multiplied version, and the subtraction. The method marks live in those lines, so a correct fraction with no working scores nothing, while correct working with a slip in the final cancelling still scores most of the marks.

Frequently asked questions

Are recurring decimals on the Foundation paper?: Not the conversion. Spec ref N10 is split: working with terminating decimals and their fractions is on both tiers, while changing a recurring decimal into a fraction and back is marked as Higher tier content in 1MA1. A Foundation paper can still show a decimal in dot notation or ask which fractions recur, but it will not ask for the algebraic conversion.
How many marks is a recurring decimal question worth on Edexcel Higher?: Usually 2-3 marks. A single repeating block is normally 2-3, and a decimal with one or two non-recurring digits in front of the block is typically 3-4 because it needs two multiplications. It tends to appear once per series rather than once per paper, so treat it as a topic to bank rather than one to rely on.
Can I just divide on the calculator instead of using algebra?: Only when the question does not ask you to prove anything. Edexcel almost always writes "prove algebraically", "use algebra to show" or "show clear algebraic working" on these, and then the marks are for the x line, the multiplied line and the subtraction. A calculator display cannot show that the recurring tail cancels.
Is 0.9 recurring, the decimal 0.9999... with the 9 repeating forever, really equal to 1?: Yes, and the standard method proves it. Let x be 0.9999..., then 10x is 9.9999..., and subtracting gives 9x = 9, so x = 1. The two are different names for the same number rather than two numbers that are very close together.
How do I tell whether a fraction gives a terminating or a recurring decimal?: Cancel the fraction fully, then factorise the denominator. If the only prime factors are 2 and 5 the decimal terminates, because the denominator divides a power of 10. Any other prime factor, such as the 3 in 12 or the 7 in 14, makes the decimal recur.

Recurring Decimals - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics