2001 AMC 12 Problem 3

Below is the professionally curated solution for Problem 3 of the 2001 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 12 solutions, or check the answer key.

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Concepts:percentage linear equation

Difficulty rating: 1240

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 $x \gt 28000$ dollars. Writing the tax with both descriptions and multiplying by $100,$ $p \cdot 28000 + (p + 2)(x - 28000) = (p + 0.25)x.$

Expanding, every term containing $p$ cancels, leaving $2x - 56000 = 0.25x,$ so $1.75x = 56000$ and $x = 32000.$

Thus, the correct answer is B.

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