NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Calculating with rational numbers

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A sports equipment supplier in Perth is allocating footballs and rugby balls to 76 schools. Each school receives 52/76 of a football and 34/76 of a rugby ball.

What is the exact total number of footballs and rugby balls distributed across all schools? Express your answer as a single exact fraction.

Concept illustrations

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Two red counters for every three blue counters. Red:blue = 2:3. There are 5 counters in all, so the red fraction of the whole is 2/5.RedRedBlueBlueBlue2 red : 3 blue · 5 counters in all
Red:blue = 2:3. There are 5 counters in all, so the red fraction of the whole is 2/5.
Translate a situation into a model. With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.$4 + $2/kmDefine tariff→C = 4 + 2dBuild rule→d=3 → C=$10Interpret
With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Each school receives 52/76 of a football and 34/76 of a rugby ball. The total number of footballs and rugby balls distributed is the sum of the fractions multiplied by the number of schools, which is 76.
  2. Total footballs = 76 schools × (52/76) football per school. The 76 in the numerator and denominator cancel, leaving 52 footballs.
  3. Total rugby balls = 76 schools × (34/76) rugby ball per school. The 76 cancels, leaving 34 rugby balls.
  4. Add the total footballs and rugby balls: 52 + 34 = 86.
  5. The total number of balls distributed is 86. This is an integer, so as a fraction it is 86/1.
  6. The question asks for the exact total as a single fraction. 86/1 is already in simplest form.
  7. Check: 52/76 simplifies to 13/19, but since 76 schools are receiving 52/76 each, the total is 76 × 52/76 = 52. This confirms the cancellation is valid.
  8. Similarly, 34/76 = 17/38, but 76 × 34/76 = 34. Cancellation is valid.
  9. No rounding or approximation is needed because the result is an exact integer.
  10. The total number of footballs and rugby balls distributed is 86, expressed as the fraction 86/1.
  11. Final verification: 52 + 34 = 86. The fraction 86/1 is exact and matches the total.
  12. The exact total number of footballs and rugby balls distributed is 86/1.

Reference answer: 86/1

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