# 2023 AMC 12B Problems/Problem 2

The following problem is from both the 2023 AMC 10B #2 and 2023 AMC 12B #2, so both problems redirect to this page.

## 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 (easy)

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 $$\frac{4}{5} \cdot x \cdot \frac{43}{40} = 43$$ $$\frac{4}{5} \cdot x \cdot \frac{1}{40} = 1$$ $$\frac{1}{5} \cdot x \cdot \frac{1}{10} = 1$$ $$x = \boxed{50}$$

## Solution 2 (One-Step Equation)

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

## Solution 3

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 4

We can assign a variable $c$ to represent the original cost of the 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 5 (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 discount price will be 40, and with sales tax applied it will be 43, so the answer choice is $\boxed{\textbf{(B) }50}$.

## Video Solution (Quick and Easy!)

~Education, the Study of Everything