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

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In a community garden project in Melbourne, the gardeners plan to allocate 69% of the total area for vegetable plots and 55% of the remaining area for compost and storage. What fraction of the total area is allocated to compost and storage?

Express your answer as an exact 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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The total area of the garden is considered as 100% or 1 (as a fraction). 69% of this area is allocated to vegetable plots.
  2. Convert 69% to a decimal: 69 ÷ 100 = 0.69.
  3. Subtract the vegetable plot area from the total to find the remaining area: 1 - 0.69 = 0.31.
  4. The compost and storage area is 55% of the remaining 31% (0.31). Convert 55% to a decimal: 55 ÷ 100 = 0.55.
  5. Multiply the remaining area by the percentage allocated to compost and storage: 0.31 × 0.55.
  6. Perform the multiplication: 0.31 × 0.55 = 0.1705.
  7. Convert 0.1705 to a fraction. First write as 1705/10000.
  8. Simplify 1705/10000 by finding the greatest common divisor (GCD) of 1705 and 10000. Factor 1705: 5 × 11 × 31. Factor 10000: 2^4 × 5^4. The common factor is 5.
  9. Divide numerator and denominator by 5: 1705 ÷ 5 = 341, 10000 ÷ 5 = 2000. So, 341/2000.
  10. Check if 341 and 2000 have any common factors. 341 = 11 × 31; 2000 = 2^4 × 5^3. No common factors, so 341/2000 is in simplest form.
  11. The fraction of the total area allocated to compost and storage is 341/2000.
  12. Verify: 69% for vegetables leaves 31%. 55% of 31% is (55/100) × (31/100) = (55 × 31) / 10000 = 1705 / 10000 = 341 / 2000. Correct.

Reference answer: 341/2000

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