Competition · AMC preparation · step 4 of 4

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)

(D)

(E)

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 1/x and check whether x ＜ 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

Set up a table of choices

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

x & 1/x & x ＜ 1/x? ; -2 & -1/2 & ? ; -1 & -1 & ? ; 0 & undefined & ruled out ; 1 & 1 & ? ; 2 & 1/2 & ?

2STEP 2

Check each row

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

x & 1/x & x ＜ 1/x? ; -2 & -1/2 & yes ; -1 & -1 & no (equal) ; 0 & undef. & no ; 1 & 1 & no (equal) ; 2 & 1/2 & no

3STEP 3

Pick the only row that works

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

-2 ＜ -1/2 → (A)

Answer

-2

Double-check the winner: -2 and

-\frac{1}{2}

are both negative, and -2 is two units left of 0 while

-\frac{1}{2}

is only half a unit left. On the number line -2 really is the smaller value, so -2 ＜

-\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

\frac{1}{2}

. The pattern fits the general rule that x ＜

\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 $-\frac{1}{2}$ on the number line, while every other choice ties or loses, so the answer is (A).

Set up a table of choices
Check each row
Pick the only row that works

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Set up a table of choices

Check each row

Pick the only row that works

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