The Scale Model Budget
A class builds a scale model of a garden. The real garden is a rectangle 12 m by 8 m. The model uses a scale of 1 cm : 2 m. Students need to buy balsa wood for the model's border (perimeter) and green felt for its area. Balsa costs $0.40 per cm; felt costs $0.15 per square cm. The class has $30. One student computes the model as 12 cm by 8 cm and says the budget is plenty. Another says the scale means divide by 2, so the model is 6 cm by 4 cm.
1. Determine the correct model dimensions and show which student is right.
Read the worked answer
Scale 1 cm : 2 m means 2 real meters per 1 model cm, so divide real lengths by 2. Length: 12 ÷ 2 = 6 cm. Width: 8 ÷ 2 = 4 cm. The second student is correct. The first student ignored the scale entirely and used meters as cm. Check by unit reasoning: 12 m × (1 cm / 2 m) = 6 cm. A quick sanity check: the model should be half the number of the real dimensions in meters, so 6 by 4, not 12 by 8. Misreading a scale as "1 to 2 means halve the number but keep the unit" happens, but the unit conversion is the whole point of a scale.
2. Compute the cost of balsa and felt and decide whether $30 is enough.
Read the worked answer
Perimeter = 2(6 + 4) = 20 cm. Balsa cost = 20 × $0.40 = $8.00. Area = 6 × 4 = 24 cm². Felt cost = 24 × $0.15 = $3.60. Total = $8.00 + $3.60 = $11.60, well under $30. Note the trap: if a student used the wrong 12-by-8 model, perimeter = 40 cm (cost $16) and area = 96 cm² (cost $14.40), total $30.40 - just over budget. So the scale error flips the answer from comfortable to impossible. This is why checking units before computing cost matters: a small scale mistake changes the conclusion, not just the number.