2012 AMC 10B Problem 6

1. Try real numbers first. Say $x = 5$ and $y = 3$, so the true value of $x - y$ is $2$.
2. Round $x$ up by $0.1$ to get $5.1$. Round $y$ down by $0.1$ to get $2.9$.
3. Her estimate is $5.1 - 2.9 = 2.2$. That is bigger than $2$ by $0.2$.
4. Now see why this always happens. Call the rounding amount $e$ (a small positive number). She uses $x + e$ and $y - e$, so her estimate is
$$(x + e) - (y - e) = x + e - y + e = x - y + 2e.$$
5. Since $e$ is positive, $2e$ is positive, so the estimate is always bigger than $x - y$, no matter what $x$ and $y$ are.

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

A difference gets bigger when the first number grows, and it also gets bigger when the number being subtracted shrinks. Here both changes push the same way, so their effects add instead of cancelling. When a word problem talks about rounding, write the rounded numbers as "number plus a little" or "number minus a little" and simplify.

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