Problem 4 · 1996 AJHSME
Medium
Fractions, Decimals & Percents
factor-common
What is the value of 2 + 4 + 6 + … + 343 + 6 + 9 + … + 51 ?
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Answer: B — 2/3.
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Hint 1 of 2
Don't add anything yet. Look at the top: 2, 4, 6, … are all even — they're 2 times 1, 2, 3, …. The bottom 3, 6, 9, … are 3 times 1, 2, 3, …. The same list is hiding in both.
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Hint 2 of 2
Factor the common multiple out of each: top = 2 × (1 + 2 + … + 17), bottom = 3 × (same sum). When the same thing sits top and bottom, it cancels — so don't waste time computing it.
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Approach: factor out the shared sum so it cancels
- Each top term is 2 × something and each bottom term is 3 × something, with the same 'somethings' (1 through 17). So top = 2(1 + 2 + … + 17) and bottom = 3(1 + 2 + … + 17).
- The big bracket appears in both, so it cancels — leaving just 2/3. The actual value of 1 + … + 17 never mattered.
- Why this transfers: before grinding out a sum or product in a fraction, hunt for a common factor on top and bottom. Cancelling first turns scary arithmetic into a one-line answer.
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