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

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In a sustainable architecture project in Perth, a designer creates a composite material using recycled glass and natural resin in a ratio of 7 : 12 by mass. The total mass of the composite used in a structural panel is 540 kg.

What is the exact mass, in kilograms, of the recycled glass used? Express your answer as an exact fraction or whole number.

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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. The ratio of recycled glass to natural resin is 7 : 12 by mass, meaning for every 7 parts of glass, there are 12 parts of resin.
  2. The total number of parts in the composite is 7 + 12 = 19 parts.
  3. The total mass of the composite is 540 kg, which corresponds to 19 parts.
  4. To find the mass of one part, divide the total mass by the total number of parts: 540 kg ÷ 19.
  5. Calculate: 540 ÷ 19 = 28.42105263... kg per part. This is an exact fraction: 540/19 kg per part.
  6. The mass of recycled glass is 7 parts, so multiply the mass per part by 7: (540/19) × 7.
  7. Perform the multiplication: (540 × 7) / 19 = 3780 / 19.
  8. Divide 3780 by 19: 19 × 198 = 3762, remainder 18. So 3780 ÷ 19 = 198 and 18/19.
  9. The result is 198 18/19 kg, but as an improper fraction: 3780/19 kg.
  10. Simplify 3780/19: 19 is prime and does not divide 3780 evenly, so the fraction is already in simplest form.
  11. The mass of recycled glass is exactly 3780/19 kg.
  12. The question asks for the exact mass in kilograms, so the answer is 3780/19 kg, which is approximately 198.947 kg, but the exact value is required.

Reference answer: 3780/19

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