Problem:
In order to estimate the value of where and are real numbers with , Xiaoli rounded up by a small amount, rounded down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
Answer Choices:
A. Her estimate is larger than .
B. Her estimate is smaller than .
C. Her estimate equals .
D. Her estimate equals .
E. Her estimate is .
Solution:
Consider and as points on the real number line, with necessarily to the right of . Then is the distance between and . Xiaoli's rounding moved to the right and moved to the left. Therefore the distance between them increased, and .
To see that the other answer choices are not correct, let and , and round each by . Then and Xiaoli's estimated difference is .
The problems on this page are the property of the MAA's American Mathematics Competitions