AMC 8 · 1999 · #3

Grade 7 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search
fraction-arithmeticsystematic-enumerationmental-arithmetic systematic-enumeration ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic
📏 Medium solution 💡 2 insights
Problem
Five triplets of numbers are listed. Four of them have a sum of 1. Find the one triplet whose sum is NOT equal to 1.

Pick an answer.

(A)
(1/2,1/3,1/6)
(B)
(2,-2,1)
(C)
(0.1,0.3,0.6)
(D)
(1.1,-2.1,1.0)
(E)
(-3/2,-5/2,5)

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Eliminate Possibilities

The question hands you five labeled candidates and asks which one fails a single, easy-to-check property (sum equals 1). That is exactly Tool #3 (Eliminate Possibilities): walk through the candidates, compute the sum of each, and cross off any whose sum is 1. The one that survives the elimination is the answer. No algebra or pattern-hunting is needed — every triplet is just three numbers to add.

1STEP 1

Test (A): the three fractions add to 1, so cross it off.

12\frac{1}{2} + 13\frac{1}{3} + 16\frac{1}{6} = 36\frac{3}{6} + 26\frac{2}{6} + 16\frac{1}{6} = 66\frac{6}{6} = 1. Sum = 1, so (A) is eliminated.
2STEP 2

Test (B): the signed integers add to 1, so cross it off.

2 + (-2) + 1 = 0 + 1 = 1. Sum = 1, so (B) is eliminated.
3STEP 3

Test (C): the three decimals add to 1, so cross it off.

0.1 + 0.3 + 0.6 = 1.0. Sum = 1, so (C) is eliminated.
4STEP 4

Test (D): the signed decimals add to 0, not 1, so (D) survives.

1.1 + (-2.1) + 1.0 = -1.0 + 1.0 = 0. Sum = 0, NOT 1. (D) survives the elimination.
5STEP 5

Check (E): the last triplet also adds to 1, so (D) is the only survivor.

−32-\frac{3}{2} + (−52-\frac{5}{2}) + 5 = −82-\frac{8}{2} + 5 = -4 + 5 = 1. Sum = 1, so (E) is eliminated. The only triplet left is (D).
Answer
(1.1,-2.1,1.0)
Four sums came out to exactly 1 and one came out to 0. The problem promises exactly one triplet fails the test, so finding exactly one failure (and four passes) matches the structure perfectly. A quick second look at (D): 1.1 + 1.0 = 2.1, and 2.1 + (-2.1) = 0 — same answer either order. The unique mismatch is (D).
💡Key takeaway

When the question is "which one is different?", just test each candidate and cross off the ones that pass. The single triplet that fails is your answer — here, (D) sums to 0, not 1.

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