2012 AMC 12B Problem 6

Below is the professionally curated solution for Problem 6 of the 2012 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 12B solutions, or check the answer key.

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Concepts:estimation inequality

Difficulty rating: 1130

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x \gt y \gt 0,$ Xiaoli 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?

Her estimate is larger than $x-y.$

Her estimate is smaller than $x-y.$

Her estimate equals $x-y.$

Her estimate equals $y-x.$

Her estimate is $0.$

Solution:

Let $d \gt 0$ be the small amount. Xiaoli computes $(x+d)-(y-d)=(x-y)+2d.$

Since $2d \gt 0,$ her estimate exceeds the true value $x-y.$

Thus, the correct answer is A.

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