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2007 AMC 10A Problem 7

2007 AMC 10A Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\$10500$ for both taxes. How many dollars was his inheritance?

A)
$30000$
B)
$32500$
C)
$35000$
D)
$37500$
E)
$40000$

Answer

Key insight

The state tax is 10% of the remaining 80%, so total tax is 20% + 8% = 28% of the inheritance.

Solution

Let the inheritance be $x$ dollars. Federal tax is $0.2x$ , leaving $0.8x$ . State tax is $10\%$ of that remainder, $0.1 \cdot 0.8x = 0.08x$ .

Total tax:
$0.2x + 0.08x = 0.28x = 10500 \quad\Longrightarrow\quad x = \frac{10500}{0.28} = 37500.$

Check: federal $7500$ , remainder $30000$ , state $3000$ , total $10500$ .

The answer is $\boxed{\textbf{(D)}\ 37500}$ .

Why this works

A percentage "of what is left" acts on a reduced base, so successive percentages combine by multiplication ( $0.8 \times 0.9 = 0.72$ kept, hence $0.28$ paid), not by addition. Once the effective overall rate is known, the problem is a single division.

Alternative approach

Test the choices: $37500$ gives federal tax $7500$ , then $10\%$ of $30000$ is $3000$ , total $10500$ . The trap answer $35000$ gives $7000 + 2800 = 9800$ , too small.

The trap

Adding the two rates to get 30% total tax, which gives 35000.

Common mistakes

Adding the two rates to get $30\%$ total tax, which gives $35000$ .
Taking the state tax as $10\%$ of the original inheritance rather than of the amount remaining after federal tax.

Techniques

Set up the equation/formula and compute; no special trick needed