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2008 AMC 10A Problem 8

2008 AMC 10A Problem 8AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

Heather compares the price of a new computer at two different stores. Store $A$ offers $15\%$ off the sticker price followed by a $\$90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves $\$15$ by buying the computer at store $A$ instead of store $B$ . What is the sticker price of the computer, in dollars?

A)
$750$
B)
$900$
C)
$1000$
D)
$1050$
E)
$1500$

Answer

Key insight

Write both prices in terms of the sticker price p: store A charges 0.85p minus 90, store B charges 0.75p, and A is 15 dollars cheaper.

Solution

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 $\boxed{\textbf{(A)}\ 750}$ .

Why this works

Percent discounts scale the price and rebates shift it, so each store's price is a linear function of $p$ . The two stores differ by $10\%$ of $p$ in the discount and by a flat $90$ in the rebate; the $15$ -dollar saving pins down $p$ . Reading "saves by buying at $A$ " as " $B$ minus $A$ equals $15$ " is the whole setup.

Alternative approach

Test the choices: at $p = 750$ , the $10\%$ gap between the discounts is $75$ , and the $90$ rebate beats that by exactly $15$ . For any larger $p$ the gap exceeds $75$ and the saving shrinks.

The trap

Setting the difference with the wrong sign (A costing 15 more than B), which gives an impossible or off-list answer.

Common mistakes

Setting the difference with the wrong sign (A costing 15 more than B), which gives an impossible or off-list answer.
Applying the $90$ rebate before the $15\%$ discount, which changes the arithmetic to $0.85(p - 90)$ .

Techniques

Set up the equation/formula and compute; no special trick needed