NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Integer and rational-number calculations

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A school in Brisbane orders 17 boxes of coloured pencils and 43 boxes of erasers for the new term. Each box of coloured pencils contains 24 pencils, and each box of erasers contains 12 erasers.

The school receives a discount of $1.50 per box on the erasers, but must pay an additional $0.75 per box for shipping on the coloured pencils. The total cost for the coloured pencils is $342.00, and the total cost for the erasers after the discount is $492.75.

What is the original price per box of erasers before the discount? Express your answer as an exact fraction in dollars.

Concept illustrations

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Common denominators as equal strips. The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.1/31/47/124/12 + 3/12 = 7/12
The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.
Adding five to negative three. Start at −3. Move five units right to reach 2.-5-4-3-2-1012345+5
Start at −3. Move five units right to reach 2.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The school ordered 43 boxes of erasers and received a $1.50 discount per box, so the total discount is 43 * $1.50 = $64.50.
  2. The total cost for erasers after discount is $492.75, so the original total cost before discount is $492.75 + $64.50 = $557.25.
  3. The original price per box of erasers is the total original cost divided by the number of boxes: $557.25 ÷ 43.
  4. Convert $557.25 to a fraction: $557.25 = 55725/100 = 2229/4 (after simplifying by dividing numerator and denominator by 25).
  5. Now divide 2229/4 by 43: (2229/4) ÷ 43 = 2229/(4 * 43) = 2229/172.
  6. Simplify 2229/172 by checking for common factors. First, divide 2229 by 172: 172 * 12 = 2064, 2229 - 2064 = 165. So 2229 = 172 * 12 + 165.
  7. Check if 2229 and 172 share any common factors. 172 = 2^2 * 43. 2229 ÷ 43 = 51.837... not integer. 2229 ÷ 2 = 1114.5, not integer. So GCD is 1.
  8. Therefore, 2229/172 is already in simplest form. The original price per box is 2229/172 dollars.
  9. Verify: 43 boxes at 2229/172 per box = 43 * 2229/172 = (43/172) * 2229 = (1/4) * 2229 = 2229/4 = 557.25. Add $64.50 discount: 557.25 - 64.50 = 492.75, matches given.
  10. The original price per box of erasers is 2229/172 dollars.

Reference answer: 2229/172

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