2010 AMC 10B Problems/Problem 11

Let the listed price be $(100 + p)$ , where $p > 0$

Coupon A saves us: $0.15(100+p) = (0.15p + 15)$

Coupon B saves us: $30$

Coupon C saves us: $0.25p$

Now, the condition is that A has to be greater than or equal to either B or C which give us the following inequalities:

$A \geq B \Leftrightarrow 0.15p + 15 \geq 30 \Leftrightarrow p \geq 100$

$A \geq C \Leftrightarrow 0.15p + 15 \geq 0.25p \Leftrightarrow p \leq 150$

We see here that the greatest possible value for p is $150 = y$ and the smallest is $100 = x$

The difference between $y$ and $x$ is $y - x \Leftrightarrow 150 - 100 = 50$

Our answer is:

$\boxed{\mathrm{(A)}= 50}$

Page