NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Equivalent fractions and decimals

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A sports equipment supplier in Brisbane is distributing 78 identical rugby balls and 68 identical cricket balls to schools. What is the exact decimal representation of the fraction of cricket balls in the total distribution?

Express your answer as a recurring decimal using correct notation.

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Half in three forms. 1/2 = 0.5 = 50%. The whole is divided into two equal parts.1 of 2 equal parts
1/2 = 0.5 = 50%. The whole is divided into two equal parts.
Common denominators as equal strips. The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.1/31/47/124/12 + 3/12 = 7/12
The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.

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Worked solution

  1. The total number of balls distributed is the sum of rugby balls and cricket balls: 78 + 68 = 146.
  2. The fraction of cricket balls is 68 out of 146, written as 68/146.
  3. Simplify the fraction by dividing both numerator and denominator by their greatest common divisor. First, find GCD(68, 146).
  4. Factor 68: 68 = 2 × 2 × 17. Factor 146: 146 = 2 × 73. The common factor is 2.
  5. Divide numerator and denominator by 2: 68 ÷ 2 = 34, 146 ÷ 2 = 73. So, 68/146 = 34/73.
  6. Now convert 34/73 to a decimal by performing long division: 34 ÷ 73.
  7. 34 ÷ 73: 73 goes into 340 four times (73 × 4 = 292), remainder 48.
  8. Bring down 0: 480. 73 goes into 480 six times (73 × 6 = 438), remainder 42.
  9. Bring down 0: 420. 73 goes into 420 five times (73 × 5 = 365), remainder 55.
  10. Bring down 0: 550. 73 goes into 550 seven times (73 × 7 = 511), remainder 39.
  11. Bring down 0: 390. 73 goes into 390 five times (73 × 5 = 365), remainder 25.
  12. The decimal repeats when a remainder repeats. After several steps, the remainder 25 reappears, confirming a recurring decimal. The decimal is 0.4657534246575342... with a repeating cycle of 6 digits. The exact recurring decimal is 0.(465753).

Reference answer: 0.(465753)

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