Problem 3 · 1992 AJHSME
Easy
Arithmetic & Operations
max-min
What is the largest difference that can be formed by subtracting two numbers chosen from the set {−16, −4, 0, 2, 4, 12}?
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Answer: D — 28.
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Hint 1 of 3
To make a subtraction a − b as big as possible, where should a sit on the number line, and where should b sit?
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Hint 2 of 3
Big difference = pick the largest possible front number and the smallest possible back number. The two extremes of the set do all the work.
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Hint 3 of 3
Subtracting a negative is the same as adding its positive — so the −16 actually helps you grow the answer, not shrink it.
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Approach: largest minus smallest — both extremes pull the gap wide
- A difference a − b is largest when a is as big as possible and b as small as possible. The biggest number is 12, the smallest is −16.
- So the largest difference is 12 − (−16). Subtracting a negative flips it to addition: 12 + 16 = 28.
- Why the negative helps: on the number line, 12 and −16 are the two outermost points, and the "distance" between them is the gap you're maximizing. The minus sign turning into a plus is exactly why reaching for the negative number gives a bigger answer, not a smaller one.
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