NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Proportion, rates and scale modelling

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Required skills (click to reveal)

A community garden in Melbourne is being redesigned using a scale model. The actual garden has a rectangular plot measuring 13 metres in length and 8.3 metres in width.

A scale of 1:25 is used for the model. What is the exact area, in square metres, of the model garden?

Express your answer as an exact fraction.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

A one-to-one-hundred length scale. At scale 1:100, a 3 cm drawing length represents 300 cm = 3 m. The two bars are symbolic and not shown at a 1:100 size ratio.Drawing: 3 cmReal: 300 cm = 3 m×100
At scale 1:100, a 3 cm drawing length represents 300 cm = 3 m. The two bars are symbolic and not shown at a 1:100 size ratio.
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.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The actual garden has a length of 13 m and a width of 8.3 m. The scale used is 1:25, meaning 1 unit on the model represents 25 units in real life.
  2. To find the model's length, divide the actual length by the scale factor: 13 m ÷ 25.
  3. 13 ÷ 25 = 13/25 m. This is the model's length in metres.
  4. To find the model's width, divide the actual width by the scale factor: 8.3 m ÷ 25.
  5. 8.3 ÷ 25 = 83/10 ÷ 25 = 83/(10×25) = 83/250 m. This is the model's width in metres.
  6. The area of a rectangle is length × width. So, model area = (13/25) × (83/250) square metres.
  7. Multiply numerators: 13 × 83 = 1079.
  8. Multiply denominators: 25 × 250 = 6250.
  9. So, area = 1079 / 6250 square metres.
  10. Check if the fraction can be simplified. The greatest common divisor of 1079 and 6250 is 1 (1079 is 13×83, both prime; 6250 is 2×5^5). So, 1079/6250 is in simplest form.
  11. The exact area of the model garden is 1079/6250 square metres.
  12. This matches the required format: an exact fraction in square metres.

Reference answer: 1079/6250 square metres

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