Problem 21 · 1985 AJHSME
Stretch
Fractions, Decimals & Percents
compound-percent
Mr. Green receives a 10% raise every year. His salary after four such raises has gone up by what percent?
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Answer: E — more than 45%.
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Hint 1 of 2
A raise is a MULTIPLY, not an add. Each year the salary becomes 1.10 times the year before — so four raises means ×1.10 four times over, not +10% four times. Those give different answers.
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Hint 2 of 2
Because every raise also raises the previous raises, the total beats the naive 4 × 10% = 40%. The question is whether the extra 'raise-on-raise' pushes past 45% — so you only need a rough size, not the exact figure.
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Approach: compound the raises step by step
- Start at 1.00 and multiply by 1.10 each year: after year 1, 1.10; year 2, 1.21; year 3, 1.331; year 4, ≈ 1.4641.
- That's about a 46% increase — more than 45%, so the answer is more than 45%.
- Why it beats 40%: simply adding 10% four times ignores that later raises act on an already-bigger salary. After two years alone you're at 1.21 (a 21% gain, not 20%) — that extra 1% snowballs, landing you past 45% by year four. This 'interest on interest' is the heart of compounding.
Another way — eyeball without exact arithmetic:
- Two 10% raises multiply to 1.10 × 1.10 = 1.21. Four raises = (1.21)² = 1.21 × 1.21.
- 1.21 × 1.21 is clearly above 1.21 × 1.20 = 1.452, so the gain exceeds 45% — answer is more than 45% without ever finishing the multiplication.
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