2014 AMC 10A Problem 7

"Must be true" means true for every allowed choice of numbers. So to reject a statement, one example where it fails is enough. To accept a statement, we need a reason it always works.

1. Statement I. If $x$ is smaller than $a$ and $y$ is smaller than $b$, then $x + y$ is smaller than $a + b$: you are adding a smaller number to a smaller number. This one always holds.
2. Statement II. Try $x = 1$, $a = 2$, $y = 1$, $b = 10$. The rules $x < a$ and $y < b$ hold. But $x - y = 0$ and $a - b = -8$, and $0 < -8$ is false. So II can fail.
3. Statement III. Negative numbers are the danger with products, so try them. Take $x = -2$, $a = -1$, $y = -2$, $b = -1$. The rules hold, since $-2 < -1$. But $xy = 4$ and $ab = 1$, and $4 < 1$ is false. So III can fail.
4. Statement IV. Use the same numbers. $\frac{x}{y} = \frac{-2}{-2} = 1$ and $\frac{a}{b} = \frac{-1}{-1} = 1$. Then $1 < 1$ is false. So IV can fail.

Only statement I must be true.

The answer is $\boxed{\textbf{(B)}\ 1}$.

Adding two inequalities that point the same way is always safe. Subtracting one from another, multiplying or dividing can all go wrong, depending on signs and sizes. When you need an example that breaks a statement, try negative numbers first; a pair like $-2 < -1$ is worth keeping ready.

(I) Inequalities in the same direction can be added: $x + y < a + b$. This one is always true.

(II) Subtracting reverses the effect of the second inequality. Take $x = 1, a = 2$ and $y = 1, b = 10$: then $x - y = 0$ but $a - b = -8$, so II fails.

(III) Products misbehave with negatives. Take $x = y = -2$ and $a = b = -1$: both hypotheses hold, but $xy = 4$ and $ab = 1$, so III fails.

(IV) The same example gives $\frac{x}{y} = 1$ and $\frac{a}{b} = 1$, so the strict inequality fails.

Exactly one statement must be true.

The answer is $\boxed{\textbf{(B)}\ 1}$.