2001 AMC 10 Problem 9

Question

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

Let her income be

x\gt28000

and write the tax two different ways

\dfrac{p}{100}(28000)

+\dfrac{p+2}{100}(x-28000)

=\dfrac{p+0.25}{100}x

Let her income be

x\ge28000.

Then

\begin{aligned} &\tfrac{p}{100}(28000) \\ &\quad {}+\tfrac{p+2}{100}(x-28000) \\ &\quad =\tfrac{p+0.25}{100}x. \end{aligned}

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.

Problem 9 in other years