2009 AMC 10B Problem 8

Below is the professionally curated solution for Problem 8 of the 2009 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 10B solutions, or check the answer key.

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Concepts:percentage algebraic manipulation

Difficulty rating: 1170

In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x?$

$12$

$17$

$20$

$25$

$35$

Solution:

Let $p$ be the starting price. After March the price is $(1.2)(0.8)(1.25)p=1.2p.$ The April drop must return it to $p,$ so it removes $0.2p,$ a fraction $x=100\cdot\dfrac{0.2p}{1.2p}=\dfrac{100}{6}\approx16.7.$

To the nearest integer, $x=17.$

Thus, the correct answer is B.

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