2023 AMC 12B Problem 2

Below is the professionally curated solution for Problem 2 of the 2023 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12B solutions, or check the answer key.

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Concepts:percentage work backwards

Difficulty rating: 1020

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?

$46$

$50$

$48$

$47$

$49$

Solution:

The final cost of a pair with original price $p$ is $0.8\times 1.075\times p=0.86p.$ Setting $0.86p\le 43$ gives $p\le 50,$ so the most expensive affordable pair originally cost $50$ dollars.

Thus, the correct answer is B.

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