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2012 AMC 10B Problem 6

2012 AMC 10B Problem 6AMC 10 Step by Step

Name: Date: September 13, 2026

Absolute Value & Inequalities · difficulty 1 of 5 · about 1 min · position P1-10

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0$ , Xiaoxi rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

A)
Her estimate is larger than $x-y
B)
Her estimate is smaller than $x-y
C)
Her estimate equals $x-y
D)
Her estimate equals $y - x
E)
Her estimate is $0

Answer

Key insight

Rounding x up and y down both push the difference the same way, so the estimate is x - y + 2e, strictly larger.

Solution

Let $e > 0$ be the small amount used for rounding. Xiaoxi replaces $x$ by $x + e$ and $y$ by $y - e$ , so her estimate is
$(x + e) - (y - e) = x - y + 2e.$
Since $e > 0$ , this exceeds $x - y$ by the positive amount $2e$ , regardless of the actual values of $x$ and $y$ .

The answer is $\boxed{\textbf{(A)}\ \text{Her estimate is larger than } x-y}$ .

Why this works

A difference grows when the first term grows or when the subtracted term shrinks. Both roundings here move in the "grow" direction, so their effects add rather than cancel. Whenever a word problem describes rounding, translate it into $\pm e$ and simplify; the sign of the leftover term settles the comparison.

Alternative approach

Test numbers: $x = 5$ , $y = 3$ , rounding amount $0.1$ . The estimate is $5.1 - 2.9 = 2.2$ , larger than the true value $2$ . Since the question asks what is necessarily true, one example rules out (B), (C), (D) and (E), which all fail here.

The trap

Thinking an upward error and a downward error cancel, when subtracting a smaller y actually adds to the difference.

Common mistakes

Thinking an upward error and a downward error cancel, when subtracting a smaller y actually adds to the difference.
Assuming the errors cancel because they are "the same amount," and choosing (C).

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer