2012 AMC 12B Problem 6

Attempt Problem 6 of the 2012 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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.

Write the rounded values as

x+d

and

y-d

for a small

d \gt 0

Subtract to compare

(x+d)-(y-d)

with

x-y

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.

Problem 6 in other years