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>
  
==Solution 1==
+
==Solution 1 (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
  
==Solution 2 (Easy)==
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==Solution 2==
 
 
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
 
 
 
==Solution 3==
 
  
 
Let the original price be <math>x</math> dollars.  
 
Let the original price be <math>x</math> dollars.  
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  ~ Minor tweak:Multpi12
 
  ~ Minor tweak:Multpi12
  
==Solution 4==
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==Solution 3==
 
We can assign a variable <math>c</math> to represent the original cost of the running 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 running 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
  
==Solution 5 (Intuition and Guessing)==
+
==Solution 4 (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 <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 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>.
  

Revision as of 20:43, 15 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

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 3

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 4 (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) }$50}$.

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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