2023 AMC 12B Problems/Problem 2

Problem

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $$43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

$\textbf{(A) }$46\qquad\textbf{(B) }$50\qquad\textbf{(C) }$48\qquad\textbf{(D) }$47\qquad\textbf{(E) }$49$

Solution 1

We can create the equation: $0.8x \cdot 1.075 = 43$ using the information given. This is because x, the original price, got reduced by 20%, or multiplied by 0.8, and it also got multiplied by 1.075 on the discounted price. Solving that equation, we get $\frac{4}{5} \cdot x \cdot \frac{43}{40} = 43$ $\frac{4}{5} \cdot x \cdot \frac{1}{40} = 1$ $\frac{1}{5} \cdot x \cdot \frac{1}{10} = 1$ $x = \boxed{50}$

~lprado

Solution 2 (Easy)

The discounted shoe is $20\%$ off the original price. So that means $1 - 0.2 = 0.8$ . There is also a $7.5\%$ sales tax charge, so $0.8 * 1.075 = 0.86$ . Now we can set up the equation $0.86x = 43$ , and solving that we get $x=\boxed{\textbf{(B) }50}$ ~ kabbybear

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2023 AMC 12B Problems/Problem 2

Problem

Solution 1

Solution 2 (Easy)