Problem 1 · 1985 AJHSME
Easy
Fractions, Decimals & Percents
cross-cancel
(3 × 5)⁄(9 × 11) × (7 × 9 × 11)⁄(3 × 5 × 7) =
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Answer: A — 1.
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Hint 1 of 2
Before multiplying anything out, scan the WHOLE expression as one big fraction: list every factor on top and every factor on the bottom. Notice anything?
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Hint 2 of 2
When every factor upstairs is matched by an identical factor downstairs, the product is forced to be 1 — you never need the actual numbers. This is the cancel-matching-factors habit: rearrange into one fraction first, then strike pairs.
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Approach: cancel matching factors
- Write it as one fraction: the top has 3, 5, 7, 9, 11 and the bottom has 9, 11, 3, 5, 7 — the SAME five numbers. Every factor on top is matched by a copy on the bottom.
- Matched pairs each make 1, so the whole product collapses to 1.
- Why this transfers: whenever a product of fractions has identical numerators and denominators (just shuffled), the answer is 1 — spotting the matching set beats computing 945 ÷ 945.
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