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

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In a community garden in Adelaide, two gardeners are planting seedlings. Gardener A plants 94 seedlings in 65 minutes.

Gardener B plants 65 seedlings in 94 minutes.

What is the exact rate of seedlings planted per minute by Gardener A, expressed as a simplified fraction?

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.
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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Gardener A plants 94 seedlings in 65 minutes. The rate of planting is seedlings per minute, which is calculated by dividing the number of seedlings by the time in minutes.
  2. The rate is given by the formula: rate = total seedlings / total time. Substitute the values: rate = 94 / 65.
  3. The fraction 94/65 must be simplified to its lowest terms. Find the greatest common divisor (GCD) of 94 and 65.
  4. Factor 94: 94 = 2 × 47. Factor 65: 65 = 5 × 13. The prime factors of 94 are 2 and 47; for 65 they are 5 and 13. There are no common prime factors.
  5. Since 94 and 65 share no common factors other than 1, the fraction 94/65 is already in its simplest form.
  6. The rate of seedlings planted per minute by Gardener A is therefore 94/65 seedlings per minute.
  7. Verify the fraction by checking if 94 ÷ 65 gives a value less than 2 (since 65 × 1 = 65, 65 × 2 = 130 > 94). The result is approximately 1.446, which is reasonable.
  8. Check that no common factor exists by testing division: 94 ÷ 13 ≈ 7.23 (not integer), 94 ÷ 5 = 18.8 (not integer), 94 ÷ 2 = 47 (but 65 is odd, so not divisible by 2).
  9. Confirm that 94 and 65 are coprime: GCD(94, 65) = 1, as confirmed by Euclidean algorithm: 94 = 1×65 + 29; 65 = 2×29 + 7; 29 = 4×7 + 1; 7 = 7×1 + 0. GCD is 1.
  10. The simplified fraction 94/65 cannot be reduced further. Therefore, the rate is exactly 94/65 seedlings per minute.
  11. The final answer is the exact rate: 94/65 seedlings per minute.
  12. The question asks for the rate as a simplified fraction. The fraction 94/65 is already simplified and correctly represents the rate.

Reference answer: 94/65 seedlings per minute

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