Difference between revisions of "2023 AMC 12B Problems/Problem 2"
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+ | {{duplicate|[[2023 AMC 10B Problems/Problem 2|2023 AMC 10B #2]] and [[2023 AMC 12B Problems/Problem 2|2023 AMC 12B #2]]}} | ||
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==Problem== | ==Problem== | ||
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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? | 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? | ||
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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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+ | 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> | ||
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+ | ~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>. | ||
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+ | ~vsinghminhas | ||
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+ | ==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>. | ||
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+ | ==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}} |
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.
Contents
[hide]Problem
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by on every pair of shoes. Carlos also knew that he had to pay a sales tax on the discounted price. He had dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
Solution 1
We can create the equation: 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
~lprado
Solution 2 (Easy)
The discounted shoe is off the original price. So that means . There is also a sales tax charge, so . Now we can set up the equation , and solving that we get ~ kabbybear
Solution 3
Let the original price be dollars. After the discount, the price becomes dollars. After tax, the price becomes dollars. So, ,
~Mintylemon66
~ Minor tweak:Multpi12
Solution 4
We can assign a variable to represent the original cost of the running shoes. Next, we set up the equation . We can solve this equation for and get .
~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 , see that the discoutn price will be 40, and with sales tax applied it will be 43, so the answer choice is .
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.