[discia] GCSE Maths › Number › Recurring Decimals to Fractions

Recurring Decimals to Fractions

discia.org 10 questions · 36 marks
Name Date
  1. Non-calculator 5 marks
    (a)

    Write the recurring decimal \(0.\dot{ 5 }\) as a fraction.

    Give your answer in its simplest form.

    (b)

    Write the recurring decimal \(0.\dot{ 6 }\dot{ 5 }\) as a fraction.

    Give your answer in its simplest form.

    Show answer
    (a) \(\frac{ 5 }{ 9 }\)
    (b) \(\frac{ 65 }{ 99 }\)
    Another like this
  2. Non-calculator 2 marks

    Write \(\frac{ 13 }{ 15 }\) as a decimal.

    Use dot notation to show which digit recurs.

    Show answer
    \(0.8\dot{ 6 }\)
    Another like this
  3. Non-calculator 3 marks

    Write the recurring decimal \(0.8\dot{ 5 }\) as a fraction.

    Give your answer in its simplest form.

    Show answer
    \(\frac{ 77 }{ 90 }\)
    Another like this
  4. Non-calculator 3 marks

    Write the recurring decimal \(0.\dot{ 1 }4\dot{ 8 }\) as a fraction in its simplest form.

    Show answer
    \(\frac{ 4 }{ 27 }\)
    Another like this
  5. Non-calculator 6 marks
    (a)

    Write \(\frac{ 7 }{9}\) as a recurring decimal.

    (b)

    Write \(\frac{ 7 }{11}\) as a recurring decimal.

    (c)

    Prove that \(0.\dot{9} = 1\)

    Show answer
    (a) \(0.\dot{ 7 }\)
    (b) \(0.\dot{ 6 }\dot{ 3 }\)
    (c) Let \(x = 0.\dot{9}\). Then \(10x = 9.\dot{9}\), so \(10x - x = 9\), giving \(9x = 9\) and \(x = 1\).
    Another like this
  6. Non-calculator 4 marks
    (a)

    Write \(2.\dot{ 8 }\dot{ 2 }\) as a mixed number.

    Give your answer in its simplest form.

    (b)

    Write \(2.\dot{ 8 }\dot{ 2 }\) as an improper fraction.

    Show answer
    (a) \(2\frac{ 82 }{ 99 }\)
    (b) \(\frac{ 280 }{ 99 }\)
    Another like this
  7. Non-calculator 4 marks

    Work out \(0.6\dot{8} + 0.\dot{6}\dot{5}\)

    Give your answer as a single fraction, not a mixed number, in its simplest form.

    Show answer
    \(\frac{ 74 }{ 55 }\)
    Another like this
  8. Non-calculator 3 marks

    Prove algebraically that \(0.9\dot{1}\dot{4}\) is equal to \(\frac{181}{198}\)

    Show answer
    Multiplying by 10 and by 1000 and subtracting gives \(990y = 905\), so \(y = \frac{ 181 }{ 198 }\), matching the printed target.
    Another like this
  9. Non-calculator 3 marks

    Here is a recurring decimal, written using only digits.

    \(0.96666\ldots\)

    Work out the repeating block, then convert the decimal to a fraction in its simplest form.

    Show answer
    \(\frac{ 29 }{ 30 }\)
    Another like this
  10. Non-calculator 3 marks

    \(\frac{ 2 }{ 27 }\) is equal to a recurring decimal.

    Work out the digit in the 58th decimal place.

    Show answer
    \(0\)
    Another like this
discia.org Recurring Decimals to Fractions · sheet 1
Answers Recurring Decimals to Fractions · sheet 1
  1. (a) \(\frac{ 5 }{ 9 }\)
    (b) \(\frac{ 65 }{ 99 }\)
  2. \(0.8\dot{ 6 }\)
  3. \(\frac{ 77 }{ 90 }\)
  4. \(\frac{ 4 }{ 27 }\)
  5. (a) \(0.\dot{ 7 }\)
    (b) \(0.\dot{ 6 }\dot{ 3 }\)
    (c) Let \(x = 0.\dot{9}\). Then \(10x = 9.\dot{9}\), so \(10x - x = 9\), giving \(9x = 9\) and \(x = 1\).
  6. (a) \(2\frac{ 82 }{ 99 }\)
    (b) \(\frac{ 280 }{ 99 }\)
  7. \(\frac{ 74 }{ 55 }\)
  8. Multiplying by 10 and by 1000 and subtracting gives \(990y = 905\), so \(y = \frac{ 181 }{ 198 }\), matching the printed target.
  9. \(\frac{ 29 }{ 30 }\)
  10. \(0\)
discia.org Recurring Decimals to Fractions · sheet 1