2001 AMC 12 Problem 3

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

dollars with

x \gt 28000,

and write the tax two different ways

Set

p \cdot 28000

+ (p + 2)(x - 28000)

= (p + 0.25)x

; the

p

terms cancel

Let her income be

x \gt 28000

dollars. Writing the tax with both descriptions and multiplying by

100,

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

Expanding, every term containing

p

cancels, leaving

2x - 56000 = 0.25x,

so

1.75x = 56000

and

x = 32000.

Thus, the correct answer is B.

Problem 3 in other years