2007 AMC 10A Problem 7

Try it first with a round number to see how the taxes work. Suppose the inheritance were $100$ dollars.

1. Federal tax: $20\%$ of $100$ is $20$. He has $80$ left.
2. State tax: $10\%$ of what is left, so $10\%$ of $80$, which is $8$.
3. Total tax: $20 + 8 = 28$ dollars out of $100$. So the taxes are always $28\%$ of the inheritance.
4. Use the real total. $28\%$ of the inheritance is $10500$:
$$0.28 \times (\text{inheritance}) = 10500, \qquad \text{inheritance} = \frac{10500}{0.28} = 37500.$$
5. Check. Federal tax: $20\%$ of $37500$ is $7500$, leaving $30000$. State tax: $10\%$ of $30000$ is $3000$. Total: $7500 + 3000 = 10500$. Correct.

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

When a percent is taken "of what is left", it acts on a smaller amount, so you cannot just add the two percents. Working through the example of $100$ dollars shows the true combined rate ($28\%$) right away. After that, one division finishes the problem.

Test the answer choices. Start with $35000$ (what you get if you wrongly add the rates): $7000$ federal, then $10\%$ of $28000$ is $2800$, total $9800$. Too small. Try $37500$: $7500 + 3000 = 10500$. That matches.

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 $\boxed{\textbf{(D)}\ 37500}$.