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

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A library in Melbourne is restocking books using two automated systems. System A can scan 46 books in 62 minutes.

What is the exact rate of books scanned per minute by System A? Express your answer 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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Worked solution

  1. The problem gives that System A scans 46 books in 62 minutes. We are to find the rate of books scanned per minute.
  2. The rate is defined as books scanned divided by time in minutes. So, rate = 46 books / 62 minutes.
  3. To express the rate as a simplified fraction, we write the ratio 46/62.
  4. We now simplify the fraction 46/62 by finding the greatest common divisor (GCD) of 46 and 62.
  5. The factors of 46 are 1, 2, 23, 46. The factors of 62 are 1, 2, 31, 62. The common factor is 2.
  6. We divide both numerator and denominator by 2: 46 ÷ 2 = 23, 62 ÷ 2 = 31.
  7. The simplified fraction is 23/31.
  8. The rate is therefore 23 books per 31 minutes, or 23/31 books per minute.
  9. We verify that 23 and 31 have no common factors (23 is prime, 31 is prime), so the fraction is fully simplified.
  10. The rate is 23/31 books per minute, which is the exact simplified fraction.
  11. We check the calculation: 23/31 ≈ 0.7419 books per minute. 0.7419 × 62 ≈ 46, confirming correctness.
  12. The exact rate of books scanned per minute by System A is 23/31.

Reference answer: 23/31 books per minute

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