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Difference between revisions of "2023 AMC 12B Problems/Problem 2"
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<math>\textbf{(A) }\$46\qquad\textbf{(B) }\$50\qquad\textbf{(C) }\$48\qquad\textbf{(D) }\$47\qquad\textbf{(E) }\$49 </math>
 
<math>\textbf{(A) }\$46\qquad\textbf{(B) }\$50\qquad\textbf{(C) }\$48\qquad\textbf{(D) }\$47\qquad\textbf{(E) }\$49 </math>
  
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==Solution 1 (easy)==
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==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:
 
We can create the equation:
 
<cmath>0.8x \cdot 1.075 = 43</cmath>
 
<cmath>0.8x \cdot 1.075 = 43</cmath>
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~lprado
 
~lprado
  
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==Solution 2 (One-Step Equation)==
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==Solution 3 (One-Step Equation)==
 
The discounted shoe is <math>20\%</math> off the original price. So that means <math>1 - 0.2 = 0.8</math>. There is also a <math>7.5\%</math> sales tax charge, so <math>0.8 * 1.075 = 0.86</math>. Now we can set up the equation <math>0.86x = 43</math>, and solving that we get <math>x=\boxed{\textbf{(B) }50}</math> ~ kabbybear
 
The discounted shoe is <math>20\%</math> off the original price. So that means <math>1 - 0.2 = 0.8</math>. There is also a <math>7.5\%</math> sales tax charge, so <math>0.8 * 1.075 = 0.86</math>. Now we can set up the equation <math>0.86x = 43</math>, and solving that we get <math>x=\boxed{\textbf{(B) }50}</math> ~ kabbybear
  
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==Solution 3==
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==Solution 4==
  
 
Let the original price be <math>x</math> dollars.  
 
Let the original price be <math>x</math> dollars.  
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After the discount, the price becomes <math> 80\%x</math> dollars.  
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After the discount, the price becomes <math> 80\%\cdot x</math> dollars.  
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After tax, the price becomes <math> 80\% \times (1+7.5\%) = 86\% x </math> dollars.
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After tax, the price becomes <math> 80\% \times (1+7.5\%) = 86\% \cdot x </math> dollars.
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So, <math>43=86\%x</math>, <math>x=\boxed{\textbf{(B) }\$50}.</math>  
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So, <math>43=86\% \cdot x</math>, <math>x=\boxed{\textbf{(B) }\$50}.</math>  
  
 
~Mintylemon66
 
~Mintylemon66
 
  ~ Minor tweak:Multpi12
 
  ~ Minor tweak:Multpi12
  
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==Solution 4==
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==Solution 5==
 
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
 
~vsinghminhas
  
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==Solution 5 (Intuition and Guessing)==
+
==Solution 6 (Intuition and Guessing)==
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We know the discount price will be 5/4, and 0.075 is equal to 3/40. So we look at answer choice <math>\textbf{(B) }</math>, see that the discoutn price will be 40, and with sales tax applied it will be 43, so the answer choice is <math>\boxed{\textbf{(B) }\$50}</math>.
+
We know the discount price will be 5/4, and 0.075 is equal to 3/40. So we look at answer choice <math>\textbf{(B) }</math>, see that the discount price will be 40, and with sales tax applied it will be 43, so the answer choice is <math>\boxed{\textbf{(B) }\$50}</math>.
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== Solution 6 (Not really a solution, DON'T DO THIS ON A REAL TEST) ==
 
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Open up a coding IDE and use Python to solve this problem. Python code:
 
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budget = 43.0
 
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discount_percentage = 20.0
 
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sales_tax_percentage = 7.5
 
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discounted_price = budget / 1.075 / 0.8
 
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print(f"${discounted_price:.2f}")
 
−
 
 
−
~Ishaan Garg
 
  
 
==Video Solution by Math-X (First understand the problem!!!)==
 
==Video Solution by Math-X (First understand the problem!!!)==
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https://www.youtube.com/watch?v=SUnhwbA5_So
 
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
  
 
==See also==
 
==See also==

Latest revision as of 10:07, 22 August 2025

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

See also

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. AMC Logo.png

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