2023 AMC 10B 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) }$47\qquad\textbf{(C) }$48\qquad\textbf{(D) }$49\qquad\textbf{(E) }$50$

Solution

Let the original price be $x$ dollars. After the discount, the price becomes $80\%x$ dollars. After tax, the price becomes $80\% \times (1+7.5\%) = 86\% x$ dollars. So, $43=86\%x$ , $x=\boxed{\textbf{(E) }$50}.$

~Mintylemon66

Solution

Original price = $\dfrac{43}{0.8 \cdot 1.075} = 50.$ That's ugly. We can sort of see that $$43$ is slightly greater than $$40$ which is 80% of $$50$ . So $50\cdot0.8\cdot1.1=44$ which is slightly greater than $$43$ , confirming $\boxed{\textbf{(E) }$50}.$

~Technodoggo

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2023 AMC 10B Problems/Problem 2

Problem

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Solution