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Difference between revisions of "2023 AMC 10B Problems/Problem 2"
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There are 2022 users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
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==Problem==
  
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Starting now, Mathbook will only allow a new friendship to be formed between two users if they have at least two friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?
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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?  
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<math>\textbf{(A) }\$46\qquad\textbf{(B) }\$47\qquad\textbf{(C) }\$48\qquad\textbf{(D) }\$49\qquad\textbf{(E) }\$50 </math>
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==Solution==
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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 tax, the price becomes <math> 80\% \times (1+7.5\%) = 86\% x </math> dollars.
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So, <math>43=86\%x</math>, <math>x=\boxed{\textbf{(E) }\$50}.</math>
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~Mintylemon66

Revision as of 14:41, 15 November 2023

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) }$47\qquad\textbf{(C) }$48\qquad\textbf{(D) }$49\qquad\textbf{(E) }$50$

Solution

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{(E) }$50}.$


~Mintylemon66

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