AMC 8 · 2000 · #2

Grade 6 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
fraction-arithmeticsystematic-enumerationparity systematic-enumerationcaseworkguess-and-check ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 2 insights
Problem
Among the five numbers -2, -1, 0, 1, 2, find the one that is strictly less than its own reciprocal.

Pick an answer.

(A)
-2
(B)
-1
(C)
0
(D)
1
(E)
2

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Guess and Check

There are only five candidates, so Tool #6 (Guess and Check) on the answer choices is the most direct path: for each x, compute 1x\frac{1}{x} and check whether x ＜ 1x\frac{1}{x}. Tool #2 (Make a List) keeps the five checks organized in a small table so nothing is missed and the lone "yes" is easy to spot. No algebra or case-by-sign reasoning is needed at this scale.

1STEP 1

List each of the five numbers, write its reciprocal 1x\frac{1}{x} beside it, and mark whether x ＜ 1x\frac{1}{x}; the x = 0 row is ruled out at once.

x & 1x\frac{1}{x} & x ＜ 1x\frac{1}{x}? ; -2 & −12-\frac{1}{2} & ? ; -1 & -1 & ? ; 0 & undefined & ruled out ; 1 & 1 & ? ; 2 & 12\frac{1}{2} & ?
2STEP 2

Check row by row: -2 sits left of −12-\frac{1}{2} so -2 ＜ −12-\frac{1}{2} is true; -1 and 1 equal their reciprocals, and 2 ＞ 12\frac{1}{2}, so those fail.

x & 1x\frac{1}{x} & x ＜ 1x\frac{1}{x}? ; -2 & −12-\frac{1}{2} & yes ; -1 & -1 & no (equal) ; 0 & undef. & no ; 1 & 1 & no (equal) ; 2 & 12\frac{1}{2} & no
3STEP 3

Exactly one row says "yes": x = -2, which is choice (A).

-2 ＜ −12-\frac{1}{2} → (A)
Answer
-2
Double-check the winner: -2 and −12-\frac{1}{2} are both negative, and -2 is two units left of 0 while −12-\frac{1}{2} is only half a unit left. On the number line -2 really is the smaller value, so -2 ＜ −12-\frac{1}{2} holds. The three "no" rows also check out: -1 = -1 and 1 = 1 fail strict less-than, and 2 is clearly bigger than 12\frac{1}{2}. The pattern fits the general rule that x ＜ 1x\frac{1}{x} requires either x ＜ -1 or 0 ＜ x ＜ 1, and only -2 on the list falls in those regions.
💡Key takeaway

With only five numbers to try, just list each one beside its reciprocal and ask "is the number the smaller one?" Here -2 beats −12-\frac{1}{2} on the number line, while every other choice ties or loses, so the answer is (A).

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