Write the recurring decimal 0.4444 using dot notation.

Find the block that repeats: 0.4444 Place the dots: 0.4 Write the answer: 0.4 Answer: 0.4

Write (4)/(9) as a recurring decimal, using dot notation.

Set up the division: 4 ÷ 9 Divide and spot the repeat: 4 ÷ 9 = 0.4444444 Write with dot notation: 0.4 Answer: 0.4

Show that 0.3 = (1)/(3).

Call the decimal x: x = 0.3333 Multiply by 10: 10x = 3.3333 Subtract to remove the tail: 10x - x = 3.333 - 0.333 Simplify both sides: 9x = 3 Solve for x: x = (3)/(9) = (1)/(3) Conclude the proof: 0.3 = (1)/(3) Answer: (1)/(3)

Convert 0.83 to a fraction in its simplest form.

Sort the fixed digits from the repeating digits: 8\;fixed,\;3\;repeats Call the decimal x: x = 0.8333333 Multiply by 10: 10x = 8.3 Multiply by 100: 100x = 83.3 Subtract to remove the tail: 100x - 10x = 83 - 8 Simplify both sides: 90x = 75 Solve for x: x = (75)/(90) Simplify the fraction: (75)/(90) = (5)/(6) Check by dividing back: 5 ÷ 6 = 0.8333333 State the answer: (5)/(6) Answer: (5)/(6)

Work out 0.6 + 0.3. Give your answer as an exact whole number.

Plan the calculation: 0.6 + 0.3 Convert 0.6666...: name it x: x = 0.6666666 Multiply x by 10: 10x = 6.6 Subtract and solve for x: 9x = 6 x = (2)/(3) Convert 0.3333...: name it y: y = 0.3333333 Multiply y by 10: 10y = 3.3 Subtract and solve for y: 9y = 3 y = (1)/(3) Add the two fractions: (2)/(3) + (1)/(3) Add the numerators: (2+1)/(3) = (3)/(3) Simplify to a whole number: (3)/(3) = 1 Check directly by adding digits: 0.6 + 0.3 = 0.9 Recall that 0.9 recurring = 1: 0.9 = 1 Confirm the two methods match: (2)/(3) + (1)/(3) = 1 = 0.9 Sense check with decimals: 0.666 + 0.333 1 State the answer: 1 Answer: 1

GCSE Recurring decimals Practice Questions

Free GCSE Recurring decimals practice questions with full step-by-step worked solutions. Covers recurring decimals, dot notation, single-digit recurring, ninths. Practise exam-style problems and check your method.

recurring decimalsdot notationsingle-digit recurringninthsfraction to decimaldivision

GCSE Higher70 questionsStep-by-step solutions

Question 1

1 markeasy

Write the recurring decimal

0.4444\ldots

using dot notation.

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Worked solution

Find the block that repeats
$0.4444\ldots$
Look along the decimal for the group of digits that keeps repeating. Here the block $4$ repeats over and over.
Place the dots
$0.\dot{4}$
In dot notation you put a dot over the first and last digit of the repeating block (a single repeating digit just gets one dot). The dots are a short way of saying "this carries on forever".
Write the answer
$0.\dot{4}$
That is the recurring decimal written neatly with dot notation.

Answer

0.\dot{4}

Question 2

2 markseasy

Write

\frac{4}{9}

as a recurring decimal, using dot notation.

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Worked solution

Set up the division
$4 \div 9$
To turn a fraction into a decimal, divide the top by the bottom. Add a decimal point and zeros after $4$ .
Divide and spot the repeat
$4 \div 9 = 0.4444444\ldots$
Dividing gives the digit $4$ again and again, because the same remainder keeps coming back. The $4$ never stops.
Write with dot notation
$0.\dot{4}$
A digit that repeats forever is shown with a dot over it. So the exact recurring decimal is $0.4444$ ....

Answer

0.\dot{4}

Question 3

2 marksintermediate

Show that

0.\dot{3} = \frac{1}{3}

Show worked solution

Worked solution

Call the decimal x
$x = 0.3333\ldots$
Let x stand for the recurring decimal 0.3333....
Multiply by $10$
$10x = 3.3333\ldots$
One repeating digit, so multiply by $10$ to shift the point one place.
Subtract to remove the tail
$10x - x = 3.333\ldots - 0.333\ldots$
Both numbers share the endless $0.3333$ ... tail, so subtracting cancels it.
Simplify both sides
$9x = 3$
$10x - x = 9x$ , and the tails cancel to leave $3$ .
Solve for x
$x = \frac{3}{9} = \frac{1}{3}$
Divide by $9$ to get $3/9$ , which cancels to $1/3$ .
Conclude the proof
$0.\dot{3} = \frac{1}{3}$
So the recurring decimal $0.3333$ ... equals exactly one third, as required.

Answer

\frac{1}{3}

Question 4

4 markshard

Convert

0.8\dot{3}

to a fraction in its simplest form.

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Worked solution

Sort the fixed digits from the repeating digits
$8\;\text{fixed},\;3\;\text{repeats}$
Only the digit(s) under the dots repeat. Here 8 stays put and 3 repeats forever, so we need two multiplications.
Call the decimal x
$x = 0.8333333\ldots$
Give the number a name: let $x = 0.8333333$ ....
Multiply by $10$
$10x = 8.\dot{3}$
Multiplying by 10 moves the point past the 1 fixed digit(s), leaving 3 just after the point.
Multiply by $100$
$100x = 83.\dot{3}$
Multiplying by 100 moves the point past the fixed digit(s) and one full repeat, so the tails still line up.
Subtract to remove the tail
$100x - 10x = 83 - 8$
Subtracting the two lines cancels the endless $0.3333$ ... tail, because both share it exactly.
Simplify both sides
$90x = 75$
$83$ minus $8$ is $75$ , and the recurring parts cancel to leave $75$ .
Solve for x
$x = \frac{75}{90}$
Divide both sides by $90$ .
Simplify the fraction
$\frac{75}{90} = \frac{5}{6}$
Both $75$ and $90$ divide down, cancelling to $5/6$ .
Check by dividing back
$5 \div 6 = 0.8333333\ldots$
Dividing 5 by 6 returns 0.8333333..., confirming the fixed digit(s) and recurring block are right.
State the answer
$\frac{5}{6}$
So $0.8333333$ ... = $5/6$ exactly.

Answer

\frac{5}{6}

Question 5

5 markschallenging

Work out

0.\dot{6} + 0.\dot{3}

. Give your answer as an exact whole number.

Show worked solution

Worked solution

Plan the calculation
$0.\dot{6} + 0.\dot{3}$
Convert each recurring decimal to a fraction, then add. Watch for a surprise at the end.
Convert 0.6666...: name it x
$x = 0.6666666\ldots$
Let $x = 0.6666$ ....
Multiply x by 10
$10x = 6.\dot{6}$
One repeating digit, so multiply by $10$ .
Subtract and solve for x
$9x = 6 \Rightarrow x = \frac{2}{3}$
Subtracting clears the tail: $9x = 6$ , so $x = 2/3$ .
Convert 0.3333...: name it y
$y = 0.3333333\ldots$
Let $y = 0.3333$ ....
Multiply y by 10
$10y = 3.\dot{3}$
One repeating digit, so multiply by $10$ .
Subtract and solve for y
$9y = 3 \Rightarrow y = \frac{1}{3}$
Subtracting clears the tail: $9y = 3$ , so $y = 1/3$ .
Add the two fractions
$\frac{2}{3} + \frac{1}{3}$
$x = 2/3$ and $y = 1/3$ , both already in thirds.
Add the numerators
$\frac{2+1}{3} = \frac{3}{3}$
Same denominator: $2 + 1 = 3$ over $3$ .
Simplify to a whole number
$\frac{3}{3} = 1$
Three thirds make one whole. The two recurring decimals add to an exact whole number.
Check directly by adding digits
$0.\dot{6} + 0.\dot{3} = 0.\dot{9}$
Adding the recurring decimals digit by digit gives $0.9999$ … = $0.9999$ ....
Recall that $0.9$ recurring = $1$
$0.\dot{9} = 1$
$0.9999$ ... is exactly $1$ (not just close to it), so both methods agree.
Confirm the two methods match
$\frac{2}{3} + \frac{1}{3} = 1 = 0.\dot{9}$
The fraction method and the digit method give the same answer of $1$ .
Sense check with decimals
$0.666\ldots + 0.333\ldots \approx 1$
Roughly $0.666 + 0.333 = 0.999\ldots$ , which is $1$ . It looks surprising that two never-ending decimals add to a tidy whole number. ✓
State the answer
$1$
So $0.6666$ ... + $0.3333$ ... = $1$ exactly, because $0.9999$ ... = $1$ .

Answer

1

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