Difference between revisions of "2023 AMC 12B Problems/Problem 2"

(Solution 4)
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We can assign a variable <math>c</math> to represent the original cost of the shoes. Next, we set up the equation <math>80\%\cdot107.5\%\cdot c=43</math>. We can solve this equation for <math>c</math> and get <math>\boxed{\textbf{(B) }$50}</math>.
 
We can assign a variable <math>c</math> to represent the original cost of the shoes. Next, we set up the equation <math>80\%\cdot107.5\%\cdot c=43</math>. We can solve this equation for <math>c</math> and get <math>\boxed{\textbf{(B) }$50}</math>.
  
~vsinghminhas & Quintuples
+
~vsinghminhas
  
 
==Solution 5 (Intuition and Guessing)==
 
==Solution 5 (Intuition and Guessing)==

Revision as of 18:47, 16 November 2023

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

~lprado

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 discoutn price will be 40, and with sales tax applied it will be 43, so the answer choice is $\boxed{\textbf{(B) }$50}$.

Solution 6 (Not really a solution, DON'T DO THIS ON A REAL TEST)

Open up a coding IDE and use Python to solve this problem. Python code:

budget = 43.0
discount_percentage = 20.0
sales_tax_percentage = 7.5
discounted_price = budget / 1.075 / 0.8
print(f"${discounted_price:.2f}")

~Ishaan Garg

Video Solution 1 by SpreadTheMathLove

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

See also

2023 AMC 10B (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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

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