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

Solution 1

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{(B) }$50}.$

~Mintylemon66

~ Minor tweak:Multpi12

Solution 2

We can assign a variable $c$ to represent the original cost of the running shoes. Next, we set up the equation $80\%\cdot107.5\%\cdot c=43$ . We can solve this equation for $c$ and get $\boxed{\textbf{(B) }$50}$ .

~vsinghminhas

Solution 3 (Intuition and Guessing)

We know the discount price will be 5/4, and 0.075 is equal to 3/40. So we look at answer choice $\textbf{(B) }$ , see that the discoutn price will be 40, and with sales tax applied it will be 43, so the answer choice is $\boxed{\textbf{(B) }$50}$ .

~lucaswujc

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2023 AMC 10B Problems/Problem 2

Contents

Problem

Solution 1

Solution 2

Solution 3 (Intuition and Guessing)