2010 AMC 10B Problem 11

Start with a few real prices to see how the coupons compare. Savings in dollars:

At low prices A loses to B, and at high prices A loses to C. A is best somewhere in the middle. Now find the exact edges.

1. Let the price be $p$ dollars. The savings are
$$\text{A: } 0.15p, \qquad \text{B: } 30, \qquad \text{C: } 0.25(p - 100) = 0.25p - 25.$$
2. A against B. We need $0.15p \ge 30$. Divide by $0.15$: $p \ge 200$.
3. A against C. We need $0.15p \ge 0.25p - 25$. Move the $p$ terms to one side: $25 \ge 0.10p$, so $p \le 250$.
4. A must do at least as well as both coupons, so $p$ is from $200$ to $250$. The smallest price is $x = 200$ and the largest is $y = 250$.
5. So $y - x = 250 - 200 = 50$.

The answer is $\boxed{\textbf{(A)}\ 50}$.

Each coupon's savings is a simple straight-line formula in the price. A percent-off coupon grows with the price, so it passes a flat $\$30$ coupon once the price is high enough. But Coupon C takes a bigger percent (of the part above $\$100$), so it eventually catches up with A. The prices where A wins form one stretch between those two crossing points, and the question asks for its length.

Let the listed price be $p > 100$. The savings are
$$A = 0.15p,\qquad B = 30,\qquad C = 0.25(p - 100).$$

Coupon A saves at least as much as B when $0.15p \ge 30$, i.e. $p \ge 200$.

Coupon A saves at least as much as C when
$$0.15p \ge 0.25p - 25 \quad\Longrightarrow\quad 25 \ge 0.10p \quad\Longrightarrow\quad p \le 250.$$

Both conditions must hold, so the valid prices form the interval $200 \le p \le 250$. Thus $x = 200$, $y = 250$, and $y - x = 50$.

The answer is $\boxed{\textbf{(A)}\ 50}$.