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AMC 8 · 2000 · #2

Grade 6 arithmetic
Archetype Small-Domain Constrained Search
fraction-arithmeticsystematic-enumerationparity systematic-enumerationcaseworkguess-and-check ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 2 insights

Problem

Which of these numbers is less than its reciprocal?

Pick an answer.

(A)
-2
(B)
-1
(C)
0
(D)
1
(E)
2
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Toolkit + CCSS Solution

Understand

Restated: Among the five numbers $-2, -1, 0, 1, 2$, find the one that is strictly less than its own reciprocal.

Givens: The five candidate numbers are $-2, -1, 0, 1, 2$; The reciprocal of a nonzero number $x$ is $\dfrac{1}{x}$; Answer choices: (A) $-2$, (B) $-1$, (C) $0$, (D) $1$, (E) $2$

Unknowns: Which single answer choice $x$ satisfies $x < \dfrac{1}{x}$

Understand

Restated: Among the five numbers $-2, -1, 0, 1, 2$, find the one that is strictly less than its own reciprocal.

Givens: The five candidate numbers are $-2, -1, 0, 1, 2$; The reciprocal of a nonzero number $x$ is $\dfrac{1}{x}$; Answer choices: (A) $-2$, (B) $-1$, (C) $0$, (D) $1$, (E) $2$

Plan

Primary tool: #6 Guess and Check

Secondary: #2 Make a List

There are only five candidates, so Tool #6 (Guess and Check) on the answer choices is the most direct path: for each $x$, compute $\tfrac{1}{x}$ and check whether $x < \tfrac{1}{x}$. 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.

Execute — Answer: A

#2 Make a List 6.NS.A.1 Step 1
  • Set up a table with one row per answer choice.
  • For each $x$, write the reciprocal $\tfrac{1}{x}$, then decide whether $x < \tfrac{1}{x}$ is true.
  • For $x = 0$ the reciprocal does not exist, so that row is automatically ruled out.
$$\begin{array}{c|c|c} x & \tfrac{1}{x} & x < \tfrac{1}{x}? \\ \hline -2 & -\tfrac{1}{2} & ? \\ -1 & -1 & ? \\ 0 & \text{undefined} & \text{ruled out} \\ 1 & 1 & ? \\ 2 & \tfrac{1}{2} & ? \end{array}$$

💡 A Grade 6 reciprocal is just $1 \div x$. Listing every $\tfrac{1}{x}$ in one column makes the comparison column an easy left-vs-right read.

#6 Guess and Check 6.NS.C.7 Step 2
  • Check each row.
  • For $x = -2$: on the number line, $-2$ sits to the left of $-\tfrac{1}{2}$, so $-2 < -\tfrac{1}{2}$ is true.
  • For $x = -1$ and $x = 1$, the number equals its reciprocal, so strict $<$ fails.
  • For $x = 2$, $2 > \tfrac{1}{2}$, so $<$ fails.
$$\begin{array}{c|c|c} x & \tfrac{1}{x} & x < \tfrac{1}{x}? \\ \hline -2 & -\tfrac{1}{2} & \text{yes} \\ -1 & -1 & \text{no (equal)} \\ 0 & \text{undef.} & \text{no} \\ 1 & 1 & \text{no (equal)} \\ 2 & \tfrac{1}{2} & \text{no} \end{array}$$

💡 For negatives, "farther from $0$" means smaller. $-2$ is farther left than $-\tfrac{1}{2}$, so $-2$ is the smaller of the two.

#6 Guess and Check 6.EE.B.5 Step 3
  • Exactly one row says "yes": $x = -2$.
  • That is choice (A).
$$-2 < -\tfrac{1}{2} \;\Rightarrow\; \textbf{(A)}$$

💡 The inequality $x < \tfrac{1}{x}$ is true for exactly one of the listed values, so that value is the answer.

[1] #2 6.NS.A.1 Set up a table with one row per answer choice. For each $x$, write the reciproca
[2] #6 6.NS.C.7 Check each row. For $x = -2$: on the number line, $-2$ sits to the left of $-\tf
[3] #6 6.EE.B.5 Exactly one row says "yes": $x = -2$. That is choice (A).

Review

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

Alternative: Tool #5 (Find a Pattern) by sign and size: a number satisfies $x < \tfrac{1}{x}$ only when $x$ and $\tfrac{1}{x}$ have the same sign and $x$ is the more-negative or smaller-positive one. That happens when $x < -1$ or $0 < x < 1$. Scanning the five choices, only $-2$ lives in $x < -1$, so the answer is (A) without computing each reciprocal.

CCSS standards used (min grade 6)

  • 6.NS.A.1 Interpret and compute quotients of fractions; understand reciprocals via division (Writing the reciprocal $\tfrac{1}{x}$ for each candidate as $1 \div x$, and noting that $1 \div 0$ is undefined.)
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers, including negatives on the number line (Comparing $-2$ with $-\tfrac{1}{2}$ to conclude $-2 < -\tfrac{1}{2}$, the only "yes" row.)
  • 6.EE.B.5 Understand solving an inequality as answering which values from a set make it true (Treating the question as "which of the five listed $x$ makes $x < \tfrac{1}{x}$ true?" and reading off the single satisfying value.)

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

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

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