Problem 12 · 1993 AJHSME
Hard
Arithmetic & Operations
order-of-operationstrial
If each of the three operation signs +, −, × is used exactly once in one of the blanks in the expression 5 __ 4 __ 6 __ 3, then the value of the result could equal
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Answer: E — 19.
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Hint 1 of 2
To reach the largest answer (19), you want multiplication to act on a big chunk — and × beats + and − no matter where it sits, so it runs first. Where should × go to grab the most?
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Hint 2 of 2
Put × on the biggest pair, 6 and 3, so multiplication fires first and gives 18. Then the + and − only nudge that 18 a little.
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Approach: place × to maximize, then let order of operations work
- Multiplication always goes first, so think of × as picking which pair gets multiplied. To land near 19, let × hit 6 and 3: 6 × 3 = 18, a big head start.
- Now slot the remaining + and − around it: 5 − 4 + 6 × 3 = 5 − 4 + 18 = 19, with each sign used exactly once.
- Why this transfers: when signs are yours to place and you want the result big, give the × the largest factors and let it run first — order of operations does the heavy lifting. Want the result small instead? Multiply the smallest pair. Same lever, opposite direction.
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