Problem 22 · 2005 AMC 8
Medium
Fractions, Decimals & Percents
unit-price-ranking
A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy.
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Answer: E — MLS (best M, then L, then S).
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Hint 1 of 2
'Best buy' only depends on price per ounce, never on the actual prices. Since everything is relative percentages, you're free to pin down two convenient anchor numbers and let the rest follow.
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Hint 2 of 2
Pick numbers that make the percentages clean — small price = $1, large size = 10 oz — then build the others and compare dollars per ounce. Lowest $/oz wins.
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Approach: anchor convenient values, then compare $ per oz
- Only ratios matter, so set Small = $1 and Large = 10 oz. Now derive the rest from the clues.
- Small size: large is twice the small, so Small = 5 oz. Medium: 50% more than small ⇒ $1.50; 20% less than large ⇒ 8 oz. Large price: 30% more than medium ⇒ 1.5·1.30 = $1.95.
- Unit prices ($/oz): S = 1/5 = 0.200, M = 1.50/8 = 0.1875, L = 1.95/10 = 0.195.
- Cheapest per ounce to priciest: M, L, S.
- Why this transfers: when a problem gives only relative sizes and prices, the answer can't depend on your starting numbers — so anchor the ones that make the arithmetic painless. And 'best deal' always means the lowest unit price, not the lowest sticker price.
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