NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Rates and unit comparisons

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A community library in Brisbane uses two automated systems to process returned books. System X processes 49 books in 73 minutes, and System Y processes 73 books in 49 minutes.

What is the exact rate of books processed per minute by System X, expressed as a simplified fraction?

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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. System X processes 49 books in 73 minutes. The rate of books processed per minute is calculated by dividing the number of books by the time in minutes.
  2. The rate is given by the formula: rate = books / time. Substitute the given values: rate = 49 / 73.
  3. The fraction 49/73 must be simplified to its lowest terms. Check for common factors between 49 and 73.
  4. Factor 49: 49 = 7 × 7. Factor 73: 73 is a prime number and not divisible by 7.
  5. Since 73 is prime and does not divide 49, and 49 has no factor in common with 73, the fraction 49/73 is already in simplest form.
  6. The rate of books processed per minute by System X is therefore 49/73 books per minute.
  7. Verify the simplification: 49 ÷ 1 = 49, 73 ÷ 1 = 73. No common divisor greater than 1 exists.
  8. The result is a proper fraction and cannot be simplified further.
  9. The rate is expressed as books per minute, so the unit is books/min.
  10. Final answer: 49/73 books per minute.
  11. Check: 49 ÷ 73 ≈ 0.671, and 49/73 is exact. No decimal approximation is required as the question asks for a simplified fraction.
  12. The exact rate of books processed per minute by System X is 49/73.

Reference answer: 49/73 books per minute

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