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

Let the price originally be $ x $. Then, after a $20$ percent discount, the price is now $ x - \frac{1}{5}x = \frac{4}{5}x $.

From the discounted price $ \frac{4}{5}x $, we now take $ \frac{7.5}{100} $ of $ \frac{4}{5}x $ and add it to $ \frac{4}{5}x $, giving us $ \frac{4}{5}x + \left(\frac{7.5}{100}\right)\left(\frac{4}{5}x\right) = \frac{4}{5}x + \frac{30}{500}x = \frac{4}{5}x + \frac{3}{50}x = \frac{43}{50}x $.

Now we write the inequality $ \frac{43}{50}x \leq \$43 $ and multiply by $ \frac{50}{43} $ on both sides to get $ x \leq \$50 $.

We want the greatest original price, which would be $$50$ or option choice $B$.

~Pinotation

Solution 2 (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 $\frac{4}{5} \cdot x \cdot \frac{43}{40} = 43$ $\frac{4}{5} \cdot x \cdot \frac{1}{40} = 1$ $\frac{1}{5} \cdot x \cdot \frac{1}{10} = 1$ $x = \boxed{50}$

~lprado

Solution 3 (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 4

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

~Mintylemon66

~ Minor tweak:Multpi12

Solution 5

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 6 (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 by Math-X (First understand the problem!!!)

https://youtu.be/EuLkw8HFdk4?si=IheDCDn6eMjae8SD&t=285 ~Math-X

Video Solution (Quick and Easy!)

https://youtu.be/Li3znsdPT1s

~Education, the Study of Everything

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=SUnhwbA5_So

Video Solution

https://youtu.be/KxLx1gSyESU

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by Interstigation

https://youtu.be/gDnmvcOzxjg?si=cYB6uChy7Ue0UT4L

2023 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2023 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

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