2014 AMC 10B Problem 7

Start with real numbers to see the pattern, then match it to the choices.

1. An example. If $B = 100$ and $A = 150$, then $A$ is $50\%$ greater than $B$, because $50$ extra out of the original $100$ is $50\%$.
2. What we did. We found the extra amount, $A - B = 50$, and compared it to the starting amount $B = 100$: $\frac{50}{100} = 0.5$, which is $50\%$.
3. In letters. The extra amount is $A - B$, and "greater than $B$" means $B$ is the starting amount. So the percent is
$$x = 100 \times \frac{A - B}{B}.$$
4. Check the example. $100 \times \frac{150 - 100}{100} = 50$. Choice (D) would give $100 \times \frac{50}{150} = 33\frac13$, which is wrong.

The answer is $\boxed{\textbf{(A)}\ 100\left(\frac{A-B}{B}\right)}$.

Any percent increase is (amount of change) divided by (starting amount), times $100$. The words "greater than $B$" tell you that $B$ is the starting amount, so $B$ goes on the bottom. When the choices are formulas, trying simple numbers like $100$ and $150$ is a quick and reliable check.

Write the meaning as an equation. Being $x\%$ greater than $B$ means $A = B + \frac{x}{100} \cdot B$. Subtract $B$: $A - B = \frac{x}{100} \cdot B$. Divide by $B$ and multiply by $100$: $x = 100 \cdot \frac{A - B}{B}$.

"$A$ is $x\%$ greater than $B$" means $A = B\left(1 + \frac{x}{100}\right)$.

Solve for $x$: $\frac{A}{B} = 1 + \frac{x}{100}$, so $\frac{x}{100} = \frac{A}{B} - 1 = \frac{A - B}{B}$, and
$$x = 100\left(\frac{A - B}{B}\right).$$

The answer is $\boxed{\textbf{(A)}\ 100\left(\frac{A-B}{B}\right)}$.