NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Rational numbers and financial modelling

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A small business in Darwin sells handmade wooden bowls. Each bowl is sold for $24.75, and the cost to produce each bowl is $11.25.

During a local market, the business sold 64 bowls and paid $320 in stall fees. The total revenue from sales was used to cover production costs and stall fees, and the profit was calculated as a percentage of total expenses.

What was the exact percentage profit made on the total expenses? Express your answer as a simplified fraction.

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A quarter is 25 percent. One of four equal parts is 25/100. For a whole of 80, each part is 20.1 of 4 equal parts
One of four equal parts is 25/100. For a whole of 80, each part is 20.
Compare the increase with the original. A price grows from 80 to 100. The added 20 is one quarter of the original 80: a 25% increase. Bar lengths share the same scale.BeforeAfter80802020 ÷ 80 × 100% = 25%
A price grows from 80 to 100. The added 20 is one quarter of the original 80: a 25% increase. Bar lengths share the same scale.

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Worked solution

  1. The business sells each wooden bowl for $24.75 and produces each for $11.25. Total revenue from 64 bowls is calculated as 64 × 24.75.
  2. Calculate 64 × 24.75: 64 × 24.75 = 64 × (25 - 0.25) = 64×25 - 64×0.25 = 1600 - 16 = 1584.
  3. Total production cost for 64 bowls is 64 × 11.25. Calculate 64 × 11.25: 64 × 11.25 = 64 × (11 + 0.25) = 64×11 + 64×0.25 = 704 + 16 = 720.
  4. Total expenses include production costs and stall fees: 720 + 320 = 1040.
  5. Profit is total revenue minus total expenses: 1584 - 1040 = 544.
  6. Percentage profit on total expenses is (profit / total expenses) × 100. Substitute values: (544 / 1040) × 100.
  7. Simplify 544 / 1040 by dividing numerator and denominator by 16: 544 ÷ 16 = 34, 1040 ÷ 16 = 65. So 544/1040 = 34/65.
  8. Now compute (34/65) × 100 = 3400 / 65.
  9. Simplify 3400 / 65: divide numerator and denominator by 5: 680 / 13.
  10. 680 ÷ 13 = 52.307... but we need the exact fraction. 680 / 13 is already in simplest form since 13 is prime and does not divide 680 evenly.
  11. The percentage profit is 680/13 %, but the question asks for the percentage as a simplified fraction. So 680/13 % = 680/13 per 100 = 680 / (13 × 100) = 680 / 1300 = 34 / 65.
  12. The exact percentage profit as a simplified fraction is 34/65. This matches the profit-to-expense ratio, and since profit is 544 and expenses 1040, 544/1040 = 34/65, confirming the result.

Reference answer: 34/65

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