NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Proportion and scaling models

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A community garden plan is drawn at a scale of 1:25. The plan shows a rectangular plot with a length of 39 cm and a width of 21 cm.

What is the actual area of the plot in square metres? Express your answer as an exact fraction.

Rectangle measurementsLength 39 cm; width 21 cm39 cm21 cm
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Concept illustrations

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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 scale of the plan is 1:25, meaning 1 cm on the plan represents 25 cm in real life.
  2. The length on the plan is 39 cm. To find the actual length, multiply by the scale factor: 39 cm × 25 = 975 cm.
  3. The width on the plan is 21 cm. To find the actual width, multiply by the scale factor: 21 cm × 25 = 525 cm.
  4. Convert the actual length from centimetres to metres: 975 cm ÷ 100 = 9.75 m.
  5. Convert the actual width from centimetres to metres: 525 cm ÷ 100 = 5.25 m.
  6. The area of a rectangle is given by length × width. Use the actual dimensions in metres: 9.75 m × 5.25 m.
  7. Convert the decimal values to fractions: 9.75 = 39/4 and 5.25 = 21/4.
  8. Multiply the fractions: (39/4) × (21/4) = (39 × 21) / (4 × 4) = 819 / 16.
  9. Simplify the fraction 819/16. Since 819 and 16 share no common factors (16 = 2^4, 819 is odd), the fraction is already in simplest form.
  10. The actual area is 819/16 square metres.
  11. Check: 819 ÷ 16 = 51.1875, and 9.75 × 5.25 = 51.1875, confirming the calculation.
  12. The actual area of the plot is 819/16 square metres.

Reference answer: 819/16

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