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…0.4444\ldots using dot notation.
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Worked solution

  1. Find the block that repeats

    0.4444…0.4444\ldots

    Look along the decimal for the group of digits that keeps repeating. Here the block 4 repeats over and over.

  2. Place the dots

    0.4˙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".

  3. Write the answer

    0.4˙0.\dot{4}

    That is the recurring decimal written neatly with dot notation.

Answer
0.4˙0.\dot{4}
Question 2
2 markseasy
Write 49\frac{4}{9} as a recurring decimal, using dot notation.
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Worked solution

  1. Set up the division

    4÷94 \div 9

    To turn a fraction into a decimal, divide the top by the bottom. Add a decimal point and zeros after 4.

  2. Divide and spot the repeat

    4÷9=0.4444444…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.

  3. Write with dot notation

    0.4˙0.\dot{4}

    A digit that repeats forever is shown with a dot over it. So the exact recurring decimal is 0.4444....

Answer
0.4˙0.\dot{4}
Question 3
2 marksintermediate
Show that 0.3˙=130.\dot{3} = \frac{1}{3}.
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Worked solution

  1. Call the decimal x

    x=0.3333…x = 0.3333\ldots

    Let x stand for the recurring decimal 0.3333....

  2. Multiply by 10

    10x=3.3333…10x = 3.3333\ldots

    One repeating digit, so multiply by 10 to shift the point one place.

  3. Subtract to remove the tail

    10x−x=3.333…−0.333…10x - x = 3.333\ldots - 0.333\ldots

    Both numbers share the endless 0.3333... tail, so subtracting cancels it.

  4. Simplify both sides

    9x=39x = 3

    10x−x=9x10x - x = 9x, and the tails cancel to leave 3.

  5. Solve for x

    x=39=13x = \frac{3}{9} = \frac{1}{3}

    Divide by 9 to get 3/9, which cancels to 1/3.

  6. Conclude the proof

    0.3˙=130.\dot{3} = \frac{1}{3}

    So the recurring decimal 0.3333... equals exactly one third, as required.

Answer
13\frac{1}{3}
Question 4
4 markshard
Convert 0.83˙0.8\dot{3} to a fraction in its simplest form.
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Worked solution

  1. Sort the fixed digits from the repeating digits

    8  fixed,  3  repeats8\;\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.

  2. Call the decimal x

    x=0.8333333…x = 0.8333333\ldots

    Give the number a name: let x=0.8333333x = 0.8333333....

  3. Multiply by 10

    10x=8.3˙10x = 8.\dot{3}

    Multiplying by 10 moves the point past the 1 fixed digit(s), leaving 3 just after the point.

  4. Multiply by 100

    100x=83.3˙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.

  5. Subtract to remove the tail

    100x−10x=83−8100x - 10x = 83 - 8

    Subtracting the two lines cancels the endless 0.3333... tail, because both share it exactly.

  6. Simplify both sides

    90x=7590x = 75

    83 minus 8 is 75, and the recurring parts cancel to leave 75.

  7. Solve for x

    x=7590x = \frac{75}{90}

    Divide both sides by 90.

  8. Simplify the fraction

    7590=56\frac{75}{90} = \frac{5}{6}

    Both 75 and 90 divide down, cancelling to 5/6.

  9. Check by dividing back

    5÷6=0.8333333…5 \div 6 = 0.8333333\ldots

    Dividing 5 by 6 returns 0.8333333..., confirming the fixed digit(s) and recurring block are right.

  10. State the answer

    56\frac{5}{6}

    So 0.8333333... = 5/6 exactly.

Answer
56\frac{5}{6}
Question 5
5 markschallenging
Work out 0.6˙+0.3˙0.\dot{6} + 0.\dot{3}. Give your answer as an exact whole number.
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Worked solution

  1. Plan the calculation

    0.6˙+0.3˙0.\dot{6} + 0.\dot{3}

    Convert each recurring decimal to a fraction, then add. Watch for a surprise at the end.

  2. Convert 0.6666...: name it x

    x=0.6666666…x = 0.6666666\ldots

    Let x=0.6666x = 0.6666....

  3. Multiply x by 10

    10x=6.6˙10x = 6.\dot{6}

    One repeating digit, so multiply by 10.

  4. Subtract and solve for x

    9x=6⇒x=239x = 6 \Rightarrow x = \frac{2}{3}

    Subtracting clears the tail: 9x=69x = 6, so x=2/3x = 2/3.

  5. Convert 0.3333...: name it y

    y=0.3333333…y = 0.3333333\ldots

    Let y=0.3333y = 0.3333....

  6. Multiply y by 10

    10y=3.3˙10y = 3.\dot{3}

    One repeating digit, so multiply by 10.

  7. Subtract and solve for y

    9y=3⇒y=139y = 3 \Rightarrow y = \frac{1}{3}

    Subtracting clears the tail: 9y=39y = 3, so y=1/3y = 1/3.

  8. Add the two fractions

    23+13\frac{2}{3} + \frac{1}{3}

    x=2/3x = 2/3 and y=1/3y = 1/3, both already in thirds.

  9. Add the numerators

    2+13=33\frac{2+1}{3} = \frac{3}{3}

    Same denominator: 2+1=32 + 1 = 3 over 3.

  10. Simplify to a whole number

    33=1\frac{3}{3} = 1

    Three thirds make one whole. The two recurring decimals add to an exact whole number.

  11. Check directly by adding digits

    0.6˙+0.3˙=0.9˙0.\dot{6} + 0.\dot{3} = 0.\dot{9}

    Adding the recurring decimals digit by digit gives 0.9999… = 0.9999....

  12. Recall that 0.9 recurring = 1

    0.9˙=10.\dot{9} = 1

    0.9999... is exactly 1 (not just close to it), so both methods agree.

  13. Confirm the two methods match

    23+13=1=0.9˙\frac{2}{3} + \frac{1}{3} = 1 = 0.\dot{9}

    The fraction method and the digit method give the same answer of 1.

  14. Sense check with decimals

    0.666…+0.333…≈10.666\ldots + 0.333\ldots \approx 1

    Roughly 0.666+0.333=0.999…0.666 + 0.333 = 0.999\ldots, which is 1. It looks surprising that two never-ending decimals add to a tidy whole number. ✓

  15. State the answer

    11

    So 0.6666... + 0.3333... = 1 exactly, because 0.9999... = 1.

Answer
11

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