2001 AMC 10 Problem 9

Below is the professionally curated solution for Problem 9 of the 2001 AMC 10, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 10 solutions, or check the answer key.

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Concepts:percentage linear equation

Difficulty rating: 1370

The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $\$28000$ of annual income plus $(p+2)\%$ of any amount above $\$28000.$ Kristin noticed that the state income tax she paid amounted to $(p+0.25)\%$ of her annual income. What was her annual income?

$\$28000$

$\$32000$

$\$35000$

$\$42000$

$\$56000$

Solution:

Let her income be $x\ge28000.$ Then

$\tfrac{p}{100}(28000)+\tfrac{p+2}{100}(x-28000)=\tfrac{p+0.25}{100}x.$

Multiplying by $100$ and expanding, all the $p$ terms cancel, leaving $2x-56000=0.25x.$ So $1.75x=56000$ and $x=32000.$

Thus, the correct answer is B.

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