NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Geometric proofs

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Required skills (click to reveal)

A school in Canberra orders 63 identical wooden panels and 51 identical metal brackets for constructing display stands. Each stand requires 3 panels and 2 brackets.

The school uses a geometric proof to verify that the maximum number of complete stands that can be built is limited by the material that runs out first. Using the concept of congruent triangles and proportional reasoning, determine the exact number of complete stands that can be built.

Provide only the exact integer value.

Concept illustrations

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A counterexample to a universal claim. A 3 cm by 2 cm rectangle has four right angles but unequal adjacent sides. It disproves the claim that every rectangle is a square. One counterexample is enough.3 cm2 cmRectangle ≠ necessarily square
A 3 cm by 2 cm rectangle has four right angles but unequal adjacent sides. It disproves the claim that every rectangle is a square. One counterexample is enough.
Adjacent angles on a straight line. The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.65°115°65° + 115° = 180°
The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The school has 63 wooden panels and 51 metal brackets. Each display stand requires 3 panels and 2 brackets.
  2. Calculate the maximum number of stands possible based on panels: 63 panels ÷ 3 panels per stand = 21 stands.
  3. Calculate the maximum number of stands possible based on brackets: 51 brackets ÷ 2 brackets per stand = 25.5 stands.
  4. Since only complete stands can be built, the number of stands is limited by the material that runs out first. 21 < 25.5, so panels are the limiting factor.
  5. The maximum number of complete stands is therefore 21, as determined by the wooden panels.
  6. Verify using proportional reasoning: for 21 stands, 21 × 3 = 63 panels used (exactly all available).
  7. For 21 stands, 21 × 2 = 42 brackets used (42 ≤ 51, so brackets are sufficient).
  8. Check if 22 stands are possible: 22 × 3 = 66 panels needed, but only 63 available → not possible.
  9. The geometric proof concept (congruent triangles and proportional reasoning) is not required for the numerical answer, as the limiting factor is determined by simple division and comparison.
  10. The maximum number of complete stands is 21, confirmed by material availability and exact usage.
  11. Final check: 21 stands use exactly 63 panels and 42 brackets, leaving 9 brackets unused. No further stands can be built.
  12. The answer is 21, as it is the largest integer for which both panel and bracket requirements are met.

Reference answer: 21

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