Problem 19 · 1985 AJHSME
Hard
Geometry & Measurement
perimeter-scaling
If the length and width of a rectangle are each increased by 10%, then the perimeter of the rectangle is increased by
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Answer: B — 10%.
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Hint 1 of 2
Don't reach for length and width numbers — none are given, which is a clue the answer doesn't depend on them. If every side gets 10% longer, what happens to the total of the sides?
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Hint 2 of 2
Perimeter is just the sum of lengths. Stretch every length by the same factor and the sum stretches by that same factor — so a 10% increase on each side is a 10% increase on the whole perimeter. (Area is different: it would grow by 1.10 × 1.10.)
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Approach: factor out the scale factor
- New perimeter = 2(1.10L + 1.10W) = 1.10 × 2(L + W) = 1.10 × (old perimeter).
- Multiplying by 1.10 is a 10% increase — the length/width values cancel out, which is why none were needed.
- Spot the trap: 21% (choice D) is the AREA increase (1.10 × 1.10 = 1.21). Length-type quantities like perimeter scale by the factor itself; area-type quantities scale by the factor squared. Knowing which is which beats plugging in numbers.
Another way — try a concrete rectangle:
- Take a 10 × 20 rectangle: perimeter = 60. Grow sides to 11 × 22: new perimeter = 66.
- 66 ÷ 60 = 1.10, a 10% increase — and the same ratio comes out for any starting rectangle.
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