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

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A construction company is designing a staircase that complies with the Australian Building Code, which specifies that the sum of the rise (vertical height) and tread (horizontal depth) of each step must not exceed 500 mm, and the rise must be between 150 mm and 200 mm. A designer proposes a staircase with a rise of 180 mm per step.

Using direct proportion, determine the exact tread depth in millimetres that satisfies the building code constraint. The model is based on the maximum allowed sum of rise and tread, and the rise is fixed at 180 mm.

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Worked solution

  1. The Australian Building Code requires that the sum of the rise and tread of each step must not exceed 500 mm.
  2. The rise is given as 180 mm per step, which is within the allowed range of 150 mm to 200 mm.
  3. Let the tread depth be t mm. The sum of rise and tread must satisfy: 180 + t ≤ 500.
  4. To maximise the tread depth while satisfying the code, we use the maximum allowed sum: 180 + t = 500.
  5. Solve for t: t = 500 - 180.
  6. Perform the subtraction: 500 - 180 = 320.
  7. The tread depth is 320 mm.
  8. Check that the rise (180 mm) and tread (320 mm) sum to 500 mm: 180 + 320 = 500 mm.
  9. Verify that the rise is within the allowed range: 150 ≤ 180 ≤ 200, which is true.
  10. Confirm that the tread depth (320 mm) is a valid, non-negative value.
  11. The solution satisfies all constraints: sum is exactly 500 mm, rise is within limits, and tread is positive.
  12. Therefore, the exact tread depth that satisfies the building code is 320 mm.

Reference answer: 320

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