Problem 22 · 1999 AMC 8
Stretch
Ratios, Rates & Proportions
substitutionunit-rate
In a far-off land three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?
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Answer: D — 2⅔ bags.
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Hint 1 of 2
Bread is the middle-man currency. Pick one good to measure everything in — rice — and convert the bread in the fish trade into rice. Then the fish↔rice rate falls right out.
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Hint 2 of 2
Chain the rates so the middle unit (bread) cancels: fish→bread→rice. This is exactly how unit conversions work in science too.
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Approach: chain the trades through a common unit so bread cancels
- Express the fish deal in rice. 3 fish = 2 loaves, and each loaf = 4 bags, so 2 loaves = 8 bags. Thus 3 fish = 8 bags of rice.
- Divide to get one fish: 8 ÷ 3 = 2⅔ bags of rice.
- The transferable move: when trades link A→B→C, convert through the shared item so it cancels — just like converting hours→minutes→seconds. (Sanity check: a fish is worth a bit more than half a loaf, and a loaf is 4 bags, so 2-and-a-bit bags per fish feels right.)
Another way — scale to whole pieces (avoid fractions of a fish):
- To dodge thirds, work with 3 fish at once. 3 fish trade for 2 loaves, and 2 loaves = 8 bags of rice, so 3 fish ↔ 8 bags.
- Reading the ratio 3 fish : 8 bags, one fish is 8/3 = 2⅔ bags.
- Bundling to the smallest whole quantities (here 3 fish) keeps the arithmetic clean and only divides once at the very end.
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