NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Ratio modelling

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A community garden in Melbourne is being redesigned with three types of soil mix for different planting beds: compost, loam, and sand. The ratio of compost to loam to sand is 24 : 30 : 94.

If the total volume of soil mix used is 1440 litres, what is the exact volume, in litres, of loam used? Express your answer as an exact fraction or whole number.

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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Worked solution

  1. The ratio of compost : loam : sand is given as 24 : 30 : 94. This means for every 24 parts compost, there are 30 parts loam and 94 parts sand.
  2. The total number of parts in the ratio is 24 + 30 + 94 = 148 parts.
  3. The total volume of soil mix used is 1440 litres, which corresponds to 148 parts.
  4. To find the volume of one part, divide the total volume by the total number of parts: 1440 ÷ 148.
  5. Calculate 1440 ÷ 148. First, simplify the fraction: 1440/148. Divide numerator and denominator by 4: 360/37.
  6. So, one part equals 360/37 litres.
  7. The loam portion is 30 parts. Multiply 30 by the volume of one part: 30 × (360/37).
  8. Compute 30 × 360 = 10800. So, 10800/37 litres.
  9. Now divide 10800 by 37. Perform long division: 37 × 291 = 10767. 10800 - 10767 = 33. So, 10800/37 = 291 and 33/37.
  10. The exact volume of loam is 10800/37 litres. This fraction cannot be simplified further since 10800 and 37 share no common factors (37 is prime and does not divide 10800).
  11. Check: 24 parts compost = 24 × 360/37 = 8640/37. 94 parts sand = 94 × 360/37 = 33840/37. Total = (8640 + 10800 + 33840)/37 = 53280/37 = 1440. Correct.
  12. Therefore, the exact volume of loam used is 10800/37 litres.

Reference answer: 10800/37

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