Problem 13 · 1996 AJHSME
Medium
Fractions, Decimals & Percents
repeated-percent
In the fall of 1996, 800 students took part in an annual school clean-up day. The organizers expect that in each of 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants expected in the fall of 1999 is
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Answer: E — 2700.
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Hint 1 of 2
A 50% increase means you keep the whole amount AND add half again — that's multiplying by 1.5 = 3/2, not adding a fixed number. Repeated growth multiplies, so apply it once per year.
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Hint 2 of 2
From 1996 you need three jumps (to '97, '98, '99), so multiply by 3/2 three times. Use the fraction 3/2 instead of 1.5 — the powers of 2 will cancel nicely against 800.
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Approach: compound by multiplying, using fractions
- Each year multiplies by 3/2 (the original plus half again). Three years means 800 × (3/2)³ = 800 × 27/8.
- Because 800 ÷ 8 = 100, this collapses to 100 × 27 = 2700 — the fraction form dodges all the decimal arithmetic.
- Why this transfers: repeated percent changes MULTIPLY (they don't add — three +50%s is not +150%). Write each as a fraction like 3/2 and watch the denominators cancel the starting number.
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