Problem 8 · 1989 AJHSME
Medium
Fractions, Decimals & Percents
distribute
(2 × 3 × 4) (12 + 13 + 14) =
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Answer: E — 26.
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Hint 1 of 3
The front product is 2×3×4 = 24, and the fractions are halves, thirds, quarters. That's no accident — what happens when you hand 24 to each fraction?
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Hint 2 of 3
Multiplying first and distributing beats finding a common denominator: spread the outside factor across each term inside the parentheses.
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Hint 3 of 3
24 is a multiple of 2, 3, AND 4, so 24×½, 24×⅓, 24×¼ are all whole numbers — no fraction arithmetic survives.
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Approach: distribute 24 over each fraction
- First simplify the front: 2×3×4 = 24. The denominators 2, 3, 4 are exactly the factors that built 24, so distributing 24 to each fraction clears every denominator: 24×½ = 12, 24×⅓ = 8, 24×¼ = 6.
- Add the three whole numbers: 12 + 8 + 6 = 26.
- Why this transfers: a number times a sum of fractions is easiest when that number is a common multiple of the denominators — distribute it inward and the fractions vanish before you ever add. Adding ½+⅓+¼ first (a clumsier 13/12) just makes more work.
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