Question

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0,$ 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 $0$

Answer: A

Concepts:estimation inequality

Difficulty rating: 870

Small Hint:

Rounding $x$ up moves it farther right on the number line

Big Hint:

Rounding $y$ down moves it farther left

Solution:

Let the small rounding amount be $d>0.$ Xiaoli replaces $x$ by $x+d$ and $y$ by $y-d.$ Her estimate is therefore $(x+d)-(y-d)=x-y+2d,$ which is larger than $x-y.$

Thus, the correct answer is A .

2012 AMC 10B Problem 6

Answer: A

Difficulty rating: 870

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years