AMC 10 · 2007 · #7
Grade 7 algebraPick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 1/4 of the inheritance, so x must be a little under four times 10500, which already points at the larger choices.
Name the inheritance
Let x be the number of dollars in the whole inheritance, so every tax becomes a piece of x.
Give the unknown a name and the word problem turns into something you can compute with.
6.EE.B.6Introduce A VariableFederal tax and the leftover
The federal tax is 20% of x, or 0.20x, and paying it leaves the other 80%, so 0.80x is still in hand.
Taking 20% away always leaves 80%, so the rest of the money is just 0.80 of the start.
Taking twenty percent away always leaves eighty percent, so the rest is a fixed share of the start.
▸ Why?
Percents count hundredths, so what is taken and what is left add to a full hundred.
▸ Why?
The money is exactly the tax plus the remainder, so removing one names the other.
State tax on what is left
The state tax is 10% of the leftover, not of x: 0.10 × 0.80x = 0.08x.
A percent only means something once you know what it is a percent of — here it is of the 80% that remains.
7.RP.A.3Convert To AlgebraAdd the taxes into one equation
The two taxes together are $10500, so adding the expressions gives 0.28x = 10500.
Two separate taxes on the same person just stack up, so their fractions of x add together.
6.EE.B.6Introduce A VariableSolve for the inheritance
Divide both sides by 0.28: x = 10500 ÷ 0.28 = 37500, so the inheritance was $37500 — choice (D).
If a known fraction of x is 10500, undo that fraction by dividing to recover the whole.
6.EE.B.7Introduce A VariableWhen 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.
- Name the inheritance
- Federal tax and the leftover
- State tax on what is left
- Add the taxes into one equation
- Solve for the inheritance