Study Companion
Topic 03 | Number & Algebra

Fractions & recurring decimals

Year 8 core: converting between fractions and decimals, recognising terminating vs recurring decimals, and predicting which fractions recur from their denominator.

45-55 min Printable practice Answer key Challenge included
VC2M8N03
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Start here: two kinds of decimal

Type 1÷41 \div 4 into a calculator. You get 0.250.25 — it stops. Now type 1÷31 \div 3. You get 0.33333…0.33333\ldots — it never stops.

Every fraction turns into one of these two kinds of decimal:

The interesting question: can you tell which kind, just by looking at the bottom of the fraction? Yes — this topic shows how.

What you will learn

1. Fraction to decimal

To convert ab\dfrac{a}{b} to a decimal, divide aa by bb. The result is either a terminating decimal (ends) or a recurring decimal (has a block of digits that repeats forever).

Worked example E Very easy: a half

12=1÷2=0.5\dfrac{1}{2} = 1 \div 2 = 0.5. Terminates after one decimal place.

Why? Because 12=510\dfrac{1}{2} = \dfrac{5}{10}, and dividing by 1010 always gives a tidy one-digit decimal.

Worked example E Very easy: a quarter

14=1÷4=0.25\dfrac{1}{4} = 1 \div 4 = 0.25. It terminates — the division ends with no remainder.

Worked example 1 Terminating

78=7÷8\dfrac{7}{8} = 7 \div 8. Long-dividing: 7.000÷8=0.8757.000 \div 8 = 0.875. This terminates after three decimal places.

Worked example 2 Recurring

13=1÷3=0.3333…\dfrac{1}{3} = 1 \div 3 = 0.3333\ldots The digit 33 repeats forever. Write it compactly as 0.3‾0.\overline{3} (a bar over the repeating block).

Other examples:

  • 16=0.16‾\dfrac{1}{6} = 0.1\overline{6} (digit 66 repeats after a non-repeating start).
  • 17=0.142857‾\dfrac{1}{7} = 0.\overline{142857} (six-digit block 142857142857 repeats).

2. Predicting from the denominator

Write the denominator in simplest form as a product of primes. If those primes are only 22 and/or 55, the decimal terminates. If any other prime appears, it recurs.

Terminating or recurring?

Rule

Simplify the fraction first. Let bb be the denominator. Factorise bb into primes.

  • Terminates if the only prime factors of bb are 22 and/or 55.
  • Recurs otherwise.
Why this works

Decimals are powers of 10=2×510 = 2 \times 5. A fraction terminates only when its denominator divides some power of 1010, which requires only the primes 22 and 55.

Worked example 3 Predict, then verify

Will 720\dfrac{7}{20} terminate? 20=22×520 = 2^2 \times 5. Only primes 22 and 55. Yes - terminates.

Check: 720=0.35\dfrac{7}{20} = 0.35. Terminates after two decimal places. ✓

Will 512\dfrac{5}{12} terminate? 12=22×312 = 2^2 \times 3. The prime 33 appears. Recurs.

Check: 512=0.416‾\dfrac{5}{12} = 0.41\overline{6}. Recurs. ✓

3. Recurring decimal to fraction

Use a bit of algebra: call the decimal xx, multiply by a power of 1010 that shifts the repeating block one full length, subtract.

Worked example E Very easy: the 10x trick on 0.111...

Let x=0.1‾=0.1111…x = 0.\overline{1} = 0.1111\ldots

  1. Multiply by 1010: 10x=1.1111…10x = 1.1111\ldots
  2. Subtract: 10x−x=110x - x = 1, so 9x=19x = 1.
  3. x=19x = \dfrac{1}{9}.

This is the simplest case of the “multiply-then-subtract” method. Every single-digit recurring decimal works the same way.

Worked example 4 Convert 0.7777 recurring to a fraction

Let x=0.7‾x = 0.\overline{7}.

  1. Multiply by 1010 (one full block): 10x=7.7‾10 x = 7.\overline{7}.
  2. Subtract the first equation: 10x−x=7.7‾−0.7‾=710 x - x = 7.\overline{7} - 0.\overline{7} = 7.
  3. 9x=79 x = 7, so x=79x = \dfrac{7}{9}.

Check: 7÷9=0.7777…7 \div 9 = 0.7777\ldots ✓

Worked example 5 Two-digit recurring block

Let x=0.27‾=0.272727…x = 0.\overline{27} = 0.272727\ldots.

  1. The block is two digits, so multiply by 100100: 100x=27.27‾100 x = 27.\overline{27}.
  2. Subtract: 99x=2799 x = 27.
  3. x=2799=311x = \dfrac{27}{99} = \dfrac{3}{11}.

Check: 3÷11=0.272727…3 \div 11 = 0.272727\ldots ✓

Practice: Year 8 core

Fluency

Fraction to decimal

    1. Write 12\dfrac{1}{2} as a decimal.
    2. Does 15\dfrac{1}{5} terminate or recur? Write its decimal.
    3. Write 110\dfrac{1}{10} as a decimal.
    4. Write 34\dfrac{3}{4} as a decimal.
    5. Write 38\dfrac{3}{8} as a decimal.
    6. Write 1120\dfrac{11}{20} as a decimal.
    7. Write 23\dfrac{2}{3} as a decimal (use the recurring bar).
    8. Write 49\dfrac{4}{9} as a decimal.
    9. Write 711\dfrac{7}{11} as a decimal (use the bar over the repeating block).
    10. Write 56\dfrac{5}{6} as a decimal.
Fluency

Predict terminating or recurring

    1. Will 940\dfrac{9}{40} terminate? Justify.
    2. Will 715\dfrac{7}{15} terminate? Justify.
    3. Will 1350\dfrac{13}{50} terminate? Justify.
    4. Will 524\dfrac{5}{24} terminate? Justify.
    5. Will 17125\dfrac{17}{125} terminate? Justify.
Fluency

Recurring to fraction

    1. Convert 0.5‾0.\overline{5} to a fraction.
    2. Convert 0.8‾0.\overline{8} to a fraction.
    3. Convert 0.12‾0.\overline{12} to a fraction.
    4. Convert 0.45‾0.\overline{45} to a fraction (and simplify).
    5. Convert 0.123‾0.\overline{123} to a fraction (and simplify).
Reasoning

Explain and spot the mistake

    1. Dev says "19=0.1\dfrac{1}{9} = 0.1". Is Dev correct? If not, what is the correct decimal?
    2. Explain why 16\dfrac{1}{6} recurs but 15\dfrac{1}{5} terminates.
    3. Kim writes 0.9‾=0.999…0.\overline{9} = 0.999\ldots and claims this is less than 11. Is Kim correct? (Hint: try the algebra trick from Worked example 4.)
    4. Show that 325\dfrac{3}{25} terminates by writing 2525 as a power of primes.
Problem solving

Applications

    1. A carpenter needs to split a 11 m length into 77 equal parts. Is each part’s length a terminating or recurring decimal in metres? Explain.
    2. Write 17\dfrac{1}{7} as a decimal. What is the smallest integer nn such that 7∣10n−17 \mid 10^n - 1? (This is the length of the repeating block.)
    3. Convert 0.142857‾0.\overline{142857} back to a fraction. (You should recognise the answer.)
    4. A recipe says “use 13\dfrac{1}{3} cup of butter”. A measuring cup has decimal markings. Explain how the cook should round.

Challenge

Reasoning

Harder reasoning

    1. Convert 0.16‾0.1\overline{6} to a fraction. (Hint: note the non-repeating “1” at the start; multiply by a power of 1010 to clear the non-repeating part first.)
    2. Convert 0.472‾0.4\overline{72} to a fraction.
    3. Two fractions have the same decimal expansion 0.45‾0.\overline{45}. Are they equal? Explain.
    4. Without computing, decide whether 123×5×7\dfrac{1}{2^3 \times 5 \times 7} terminates. Justify.
Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Year 8 core - answers

Fluency

Fraction to decimal

    1. 0.50.5
    2. Terminates; 15=0.2\dfrac{1}{5} = 0.2.
    3. 0.10.1
    4. 0.750.75
    5. 0.3750.375
    6. 0.550.55
    7. 0.6‾0.\overline{6} (recurring - the digit 66 repeats)
    8. 0.4‾0.\overline{4}
    9. 0.63‾0.\overline{63} (the two-digit block 6363 repeats)
    10. 0.83‾0.8\overline{3} (only the 33 repeats)
Fluency

Predict terminating or recurring

    1. Terminates. 40=23×540 = 2^3 \times 5 - only primes 22 and 55.
    2. Recurs. 15=3×515 = 3 \times 5 - the prime 33 appears.
    3. Terminates. 50=2×5250 = 2 \times 5^2.
    4. Recurs. 24=23×324 = 2^3 \times 3.
    5. Terminates. 125=53125 = 5^3.
Fluency

Recurring to fraction

    1. 59\dfrac{5}{9}. Method: x=0.5‾x = 0.\overline{5}; 10x−x=510x - x = 5; 9x=59x = 5.
    2. 89\dfrac{8}{9}.
    3. 1299=433\dfrac{12}{99} = \dfrac{4}{33}. Method: two-digit block, so 100x−x=12100 x - x = 12.
    4. 4599=511\dfrac{45}{99} = \dfrac{5}{11}.
    5. 123999=41333\dfrac{123}{999} = \dfrac{41}{333}.
Reasoning

Explain and spot the mistake

    1. Wrong. 19=0.1‾=0.1111…\dfrac{1}{9} = 0.\overline{1} = 0.1111\ldots, not 0.10.1. Dev has truncated instead of noting the recurrence.
    2. 6=2×36 = 2 \times 3; the prime 33 means no power of 1010 is a multiple of 66, so the decimal must recur. 55 is itself a prime factor of 1010, so 15=0.2\dfrac{1}{5} = 0.2 terminates.
    3. Kim is wrong - 0.9‾0.\overline{9} equals 11 exactly. Let x=0.9‾x = 0.\overline{9}; 10x=9.9‾10x = 9.\overline{9}; 10x−x=910x - x = 9; 9x=99x = 9; x=1x = 1.
    4. 25=5225 = 5^2. Since the only prime is 55, 325\dfrac{3}{25} terminates. Specifically 325=12100=0.12\dfrac{3}{25} = \dfrac{12}{100} = 0.12.
Problem solving

Applications

    1. Each part is 17\dfrac{1}{7} m =0.142857‾= 0.\overline{142857} m - recurring. (77 is neither 22 nor 55.)
    2. 17=0.142857‾\dfrac{1}{7} = 0.\overline{142857}; the block has length 66, so n=6n = 6 satisfies 7∣106−1=9999997 \mid 10^6 - 1 = 999999.
    3. 142857999999=17\dfrac{142857}{999999} = \dfrac{1}{7}.
    4. 13≈0.333\dfrac{1}{3} \approx 0.333, so round to the nearest practical value of the measuring cup (e.g. if marked every 14\tfrac{1}{4} cup, round up to 13≈0.33\tfrac{1}{3} \approx 0.33 or accept 13\tfrac{1}{3} directly).

Challenge - answers

Reasoning

Harder reasoning

    1. 16\dfrac{1}{6}. Method: x=0.16‾x = 0.1\overline{6}. 10x=1.6‾10 x = 1.\overline{6}; 100x=16.6‾100 x = 16.\overline{6}; 100x−10x=90x=15100 x - 10 x = 90 x = 15; x=1590=16x = \dfrac{15}{90} = \dfrac{1}{6}.
    2. 2655\dfrac{26}{55}. Method: x=0.472‾x = 0.4\overline{72}. 10x=4.72‾10x = 4.\overline{72}; 1000x=472.72‾1000x = 472.\overline{72}; 1000x−10x=4681000x - 10x = 468; 990x=468990x = 468; x=468990=2655x = \dfrac{468}{990} = \dfrac{26}{55}.
    3. Yes, they must be equal - two numbers with the same decimal expansion are the same number. Different fractions would give different expansions.
    4. Recurs. The prime 77 is present in the denominator, which is neither 22 nor 55.

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