2001 AMC 10 Problem 9

P1-10~3 minPrint

Ratios, Percents & Averages Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

The state income tax where Kristin lives is charged 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?

Show solution without answering

Key insight

See the tax as p% on everything plus an extra 2% above $28000. The p% parts cancel, so that extra 2% equals 0.25% of her income.

Solution

1. First, her income must be more than $\$28000$ . If it were $\$28000$ or less, all of it would be taxed at exactly $p\%$ , not $(p + 0.25)\%$ .
2. Call her income $I$ . Split it into the first $\$28000$ and the extra part, $I - 28000$ .
3. The extra part is taxed at $(p + 2)\%$ , which is $p\%$ plus another $2\%$ . So her tax is the same as:
- $p\%$ on all of her income, plus
- an extra $2\%$ on the part above $\$28000$ .
4. The problem also says her tax is $(p + 0.25)\%$ of her income, which is:
- $p\%$ on all of her income, plus
- an extra $0.25\%$ on all of her income.
5. The first lines of steps 3 and 4 are the same, so the second lines must be equal too:
$2\% \text{ of } (I - 28000) = 0.25\% \text{ of } I.$
6. Multiply both sides by $100$ to remove the percent signs: $2(I - 28000) = 0.25I$ .
7. Expand: $2I - 56000 = 0.25I$ . Move the $I$ terms together: $1.75I = 56000$ .
8. Divide: $I = 56000 \div 1.75 = 32000$ .
9. Check: the extra part is $\$4000$ , and $2\%$ of $4000$ is $\$80$ . Also $0.25\%$ of $32000$ is $\$80$ . They match.

The answer is $\boxed{\textbf{(B)}\ \$32000}$ .

Why this works

A tax with two rates can always be seen as "the lower rate on everything, plus a surcharge on the amount over the cutoff." Seen that way, the unknown rate $p$ appears on both sides and cancels, so you never need to know it. When a letter you cannot find appears on both sides, look for a way to make it cancel.

Alternative approach

Test the choices using the surcharge idea from step 5: compare $2\%$ of the amount over $\$28000$ with $0.25\%$ of the income.

Income	$2\%$ of excess	$0.25\%$ of income
$\$32000$	$\$80$	$\$80$
$\$35000$	$\$140$	$\$87.50$
$\$42000$	$\$280$	$\$105$

Only $\$32000$ balances.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Let Kristin's income be $I$ dollars. If $I \le 28000$ her rate would be exactly $p\%$ , not $(p+0.25)\%$ , so $I > 28000$ . Her tax, in dollars, is
$\frac{p}{100}\cdot 28000 + \frac{p+2}{100}(I - 28000).$
Regroup: this equals $\frac{p}{100}\cdot I + \frac{2}{100}(I - 28000)$ , i.e. $p\%$ of everything plus an extra $2\%$ of the part above $28000$ .

The tax is also $\frac{p+0.25}{100} I = \frac{p}{100} I + \frac{0.25}{100} I$ . Comparing, the $p\%$ pieces cancel and
$2(I - 28000) = 0.25 I \quad\Longrightarrow\quad 1.75 I = 56000 \quad\Longrightarrow\quad I = 32000.$

The answer is $\boxed{\textbf{(B)}\ \$32000}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)