NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Rates and unit comparisons

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A library in Sydney has two staff members who process books at different rates. Staff member A can process 89 books in 75 minutes.

Staff member B can process 75 books in 89 minutes.

What is the exact rate of books processed per minute by Staff member A, 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.
Square the length conversion factor. A 1 m by 1 m square is 100 cm by 100 cm. Its area is 10,000 cm². The 10-by-10 grid shows 100 squares, each 10 cm by 10 cm.1 m = 100 cm1 m100 cm
A 1 m by 1 m square is 100 cm by 100 cm. Its area is 10,000 cm². The 10-by-10 grid shows 100 squares, each 10 cm by 10 cm.

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Worked solution

  1. Staff member A processes 89 books in 75 minutes. The rate is books per minute, so divide the number of books by the time in minutes.
  2. The rate is calculated as 89 books ÷ 75 minutes. This gives the fraction 89/75 books per minute.
  3. Check if 89 and 75 have any common factors. 89 is a prime number (it is only divisible by 1 and itself). 75 factors as 3 × 5².
  4. Since 89 is prime and does not divide 75, the fraction 89/75 cannot be simplified further.
  5. The rate of books processed per minute by Staff member A is therefore 89/75 books per minute.
  6. Verify the fraction by converting to decimal: 89 ÷ 75 = 1.18666..., which matches 89/75.
  7. No further simplification is possible because 89 and 75 share no common divisors other than 1.
  8. The exact rate is 89/75 books per minute, expressed as a simplified fraction.

Reference answer: 89/75 books per minute

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