2008 AMC 10A Problem 8

1. Call the sticker price $p$ dollars.
2. Store $A$: $15\%$ off means Heather pays $85\%$ of the price, which is $0.85p$. Then $90$ dollars come back, so she pays $0.85p - 90$.
3. Store $B$: $25\%$ off means she pays $75\%$ of the price, which is $0.75p$.
4. Buying at $A$ saves $15$ dollars, so store $B$'s price minus store $A$'s price is $15$:
$$0.75p - (0.85p - 90) = 15.$$
5. Simplify the left side: $0.75p - 0.85p + 90 = 90 - 0.10p$. So
$$90 - 0.10p = 15, \qquad 0.10p = 75, \qquad p = 750.$$
6. Check with real numbers. At store $A$: $0.85 \times 750 = 637.50$, minus $90$ is $547.50$. At store $B$: $0.75 \times 750 = 562.50$. The difference is $15$ dollars, as required.

The answer is $\boxed{\textbf{(A)}\ 750}$.

A percent discount takes off an amount that grows with the price, while a rebate takes off a fixed amount. Store $B$'s extra $10\%$ off is compared against store $A$'s fixed $90$ dollars. Writing both prices in terms of one letter turns the word problem into a one-step equation.

Think in words. Store $B$ takes off $25\% - 15\% = 10\%$ more of the price than store $A$. Store $A$ makes up for that with its $90$ dollar rebate, and still comes out $15$ dollars ahead. So the extra $10\%$ is worth $90 - 15 = 75$ dollars, and the full price is $10 \times 75 = 750$ dollars.

Let the sticker price be $p$ dollars.

Store $A$: $15\%$ off leaves $0.85p$, then the rebate takes off $90$, so Heather pays $0.85p - 90$.

Store $B$: $25\%$ off leaves $0.75p$.

Store $A$ is $15$ dollars cheaper:
$$0.75p - (0.85p - 90) = 15 \quad\Longrightarrow\quad 90 - 0.10p = 15 \quad\Longrightarrow\quad 0.10p = 75.$$
So $p = 750$.

Check: store $A$ charges $637.50 - 90 = 547.50$; store $B$ charges $562.50$; the difference is $15$.

The answer is $\boxed{\textbf{(A)}\ 750}$.