Recurring decimals Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Recurring decimals questions. See exactly how to solve problems on recurring decimals, dot notation, single-digit recurring, ninths.

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.

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 44 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
1 markeasy
Write 0.7777…0.7777\ldots as a fraction.

Worked solution

  1. Use the ninths pattern

    0.7˙=790.\dot{7} = \frac{7}{9}

    A single repeating digit over one place is always that digit over 99. So 0.77770.7777... means 77 ninths.

  2. Check it is in its simplest form

    79\frac{7}{9}

    The numerator 77 and denominator 99 share no common factor, so it cannot be cancelled down.

  3. State the answer

    79\frac{7}{9}

    That is the exact fraction equal to the recurring decimal.

Answer
79\frac{7}{9}
Question 3
1 markeasy
Write the recurring decimal 0.2˙0.\dot{2} as a fraction.

Worked solution

  1. Use the ninths pattern

    0.2˙=290.\dot{2} = \frac{2}{9}

    A single repeating digit over one place is always that digit over 99. So 0.22220.2222... means 22 ninths.

  2. Check it is in its simplest form

    29\frac{2}{9}

    The numerator 22 and denominator 99 share no common factor, so it cannot be cancelled down.

  3. State the answer

    29\frac{2}{9}

    That is the exact fraction equal to the recurring decimal.

Answer
29\frac{2}{9}
Question 4
1 markeasy
Write the recurring decimal 0.5555…0.5555\ldots using dot notation.

Worked solution

  1. Find the block that repeats

    0.5555…0.5555\ldots

    Look along the decimal for the group of digits that keeps repeating. Here the block 55 repeats over and over.

  2. Place the dots

    0.5˙0.\dot{5}

    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

    0.5˙0.\dot{5}

    That is the recurring decimal written neatly with dot notation.

Answer
0.5˙0.\dot{5}
Question 5
2 markseasy
Write 0.3˙0.\dot{3} as a fraction in its simplest form.

Worked solution

  1. Use the ninths pattern

    0.3˙=390.\dot{3} = \frac{3}{9}

    A single repeating digit over one place is that digit over 99. So start by writing 0.33330.3333... as 33 ninths.

  2. Simplify the fraction

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

    Both 33 and 99 divide by 33, so cancel down to get the simplest form.

  3. State the answer

    13\frac{1}{3}

    That is the recurring decimal written as an exact fraction in its simplest form.

Answer
13\frac{1}{3}

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