Problem 8 · 1991 AJHSME
Medium
Arithmetic & Operations
max-quotientsigns
What is the largest quotient that can be formed using two numbers chosen from the set {−24, −3, −2, 1, 2, 8}?
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Answer: D — 12.
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Hint 1 of 3
To make a quotient as LARGE as possible you want it positive and big. A division is positive when both numbers share a sign — and the set's two extreme-magnitude numbers (−24 and... ) happen to both be negative. What pairing makes the result both positive and huge?
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Hint 2 of 3
Big quotient = big top ÷ small bottom. For a positive answer the two numbers must match signs; the negatives let you do that AND use the biggest magnitude, −24.
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Hint 3 of 3
Try −24 divided by a small-magnitude negative. Which negative in the set has the smallest size?
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Approach: make it positive (matching signs) with the biggest numerator over the smallest denominator
- A quotient grows when the top is large and the bottom is small, and it's positive only when the two numbers share a sign. The biggest-magnitude number is −24 (negative), so pair it with another negative to keep the result positive.
- The smallest-magnitude negative is −2. Then −24 ÷ −2 = +12 — large top, small bottom, and the two negatives cancel to give a positive.
- So the largest quotient is 12.
- Trap to dodge: −24 ÷ −3 = 8 is smaller, and 8 ÷ 1 = 8 also loses — the winning move uses BOTH the biggest magnitude on top and a sign match. Reaching for the literal largest number (8) on top is the bait that gives only 8 ÷ ... at best.
- Why this transfers: with signed numbers, "largest result" problems split into two decisions — fix the sign first (match signs to go positive), then maximize magnitude (biggest ÷ smallest).
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