AMC 10 · 2007 · #5
Grade 7 algebraPick an answer.
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
Naming the original total gives everything one letter to hang on.
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 first tax leaves a plain fraction of that total.
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 of the money is eight tenths of the start.
▸ Why?
A percent is a count out of a hundred, so what is taken and what is left add up to a hundred.
▸ Why?
The money taken and the money left make up the whole, so naming one names the other.
State tax on what is left
The second tax is a percent of that leftover, not the original.
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
Adding the taxes gives one equation.
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
Solving gives 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