2023 AMC 12B Problems/Problem 2: Difference between revisions

Revision as of 18:52, 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

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 (Easy)

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 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 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 $\boxed{\textbf{(B) }\$50}$ .

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

@@ Line 1: / Line 1: @@
+{{duplicate|[[2023 AMC 10B Problems/Problem 2|2023 AMC 10B #2]] and [[2023 AMC 12B Problems/Problem 2|2023 AMC 12B #2]]}}
 ==Problem==
 Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by <math>20\%</math> on every pair of shoes. Carlos also knew that he had to pay a <math>7.5\%</math> sales tax on the discounted price. He had <math>\$43</math> dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
@@ Line 20: / Line 21: @@
 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.
+After the discount, the price becomes <math> 80\%x</math> dollars.
+After tax, the price becomes <math> 80\% \times (1+7.5\%) = 86\% x </math> dollars.
+So, <math>43=86\%x</math>, <math>x=\boxed{\textbf{(B) }\$50}.</math>
+~Mintylemon66
+ ~ Minor tweak:Multpi12
+==Solution 4==
+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
+==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 <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>.
+==See also==
+{{AMC10 box|year=2023|ab=B|num-b=1|num-a=3}}
+{{AMC12 box|year=2023|ab=B|num-b=1|num-a=3}}
+{{MAA Notice}}

Page

Toolbox

Search