AMC 10 · 2007 · #7
Grade 7 arithmeticLast 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 \textdollar10500 for both taxes. How many dollars was his inheritance?
(A) 30000(B) 32500(C) 35000(D) 37500(E) 40000
Pick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A man pays $20\%$ of his inheritance in federal tax, then pays $10\%$ of whatever is left in state tax. The two taxes together come to $\textdollar 10500$. Find the size of the whole inheritance in dollars.
Givens: Federal tax is $20\%$ of the full inheritance; State tax is $10\%$ of the amount left after the federal tax; The two taxes add up to $\textdollar 10500$; Answer choices: (A) $30000$, (B) $32500$, (C) $35000$, (D) $37500$, (E) $40000$
Unknowns: The dollar amount of the whole inheritance
Understand
Restated: A man pays $20\%$ of his inheritance in federal tax, then pays $10\%$ of whatever is left in state tax. The two taxes together come to $\textdollar 10500$. Find the size of the whole inheritance in dollars.
Givens: Federal tax is $20\%$ of the full inheritance; State tax is $10\%$ of the amount left after the federal tax; The two taxes add up to $\textdollar 10500$; Answer choices: (A) $30000$, (B) $32500$, (C) $35000$, (D) $37500$, (E) $40000$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
The inheritance is the one thing we do not know, so Tool #4 (Introduce a Variable) names it $x$ and lets every tax become a piece of $x$. Tool #13 (Convert to Algebra) turns the words "$20\%$ of it" and "$10\%$ of what is left" into expressions, so the two taxes add into a single equation about $x$. Tool #3 (Eliminate Possibilities) offers a fast check: the total tax is a bit more than $\tfrac14$ of the inheritance, so $x$ must be a little under four times $10500$, which already points at the larger choices.
Execute — Answer: D
6.EE.B.6 Step 1 Name the inheritance
- Let $x$ be the number of dollars in the whole inheritance.
- Every tax in the problem is a percentage of some amount, and once we have the letter $x$ we can write each of those amounts as a piece of $x$.
💡 Give the unknown a name and the word problem turns into something you can compute with.
7.RP.A.3 Step 2 Federal tax and the leftover
- The federal tax is $20\%$ of the inheritance, which is $0.20x$.
- Paying it leaves the other $80\%$, so the amount still in his hands is $0.80x$.
- This leftover is the base for the next tax.
💡 Taking $20\%$ away always leaves $80\%$, so the rest of the money is just $0.80$ of the start.
7.RP.A.3 Step 3 State tax on what is left
- The state tax is $10\%$ of the leftover, not of the original.
- So it is $10\%$ of $0.80x$, which is $0.10 \times 0.80x = 0.08x$.
- Watch the base carefully: the $10\%$ is measured against $0.80x$, giving $0.08x$, not $0.10x$.
💡 A percent only means something once you know what it is a percent of — here it is of the $80\%$ that remains.
6.EE.B.6 Step 4 Add the taxes into one equation
- The two taxes together equal $\textdollar 10500$.
- Adding the expressions gives $0.20x + 0.08x = 0.28x$, so $0.28x = 10500$.
- This is a single equation with one unknown.
💡 Two separate taxes on the same person just stack up, so their fractions of $x$ add together.
6.EE.B.7 Step 5 Solve for the inheritance
- Divide both sides by $0.28$ to undo the multiplication: $x = 10500 \div 0.28 = 37500$.
- So the inheritance was $\textdollar 37500$, which is choice (D).
💡 If a known fraction of $x$ is $10500$, undo that fraction by dividing to recover the whole.
6.EE.B.6 Let $x$ be the number of dollars in the whole inheritance. Every tax in the prob 7.RP.A.3 The federal tax is $20\%$ of the inheritance, which is $0.20x$. Paying it leaves 7.RP.A.3 The state tax is $10\%$ of the leftover, not of the original. So it is $10\%$ of 6.EE.B.6 The two taxes together equal $\textdollar 10500$. Adding the expressions gives $ 6.EE.B.7 Divide both sides by $0.28$ to undo the multiplication: $x = 10500 \div 0.28 = 3 Review
Reasonableness: Check by running the tax forward on $x = 37500$: federal is $0.20 \times 37500 = 7500$, leaving $30000$; state is $0.10 \times 30000 = 3000$. The two taxes are $7500 + 3000 = 10500$, exactly the given total. The size is sensible too: total tax is $0.28$ of the inheritance, a bit more than a quarter, so the inheritance should be a bit under $4 \times 10500 = 42000$, and $37500$ fits.
Alternative: Track the money he keeps instead of the tax he pays. He keeps $80\%$ after federal tax, then keeps $90\%$ of that after state tax, so he keeps $0.80 \times 0.90 = 0.72$ of the inheritance. That means he pays $1 - 0.72 = 0.28$ of it in tax, giving the same equation $0.28x = 10500$ and the same answer $x = 37500$.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Naming the inheritance $x$ and writing the combined-tax expression $0.28x$.)7.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems (Taking $20\%$ of $x$, finding the leftover $0.80x$, and taking $10\%$ of that leftover to get $0.08x$.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Solving $0.28x = 10500$ by dividing both sides by $0.28$ to get $x = 37500$.)
⭐ When one percent is taken and then another percent is taken from what is left, write each tax as a piece of the same unknown, add the pieces, and solve the single equation.
⭐ When one percent is taken and then another percent is taken from what is left, write each tax as a piece of the same unknown, add the pieces, and solve the single equation.
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