AMC 10 · 2008 · #6
Grade 7 algebraPick an answer.
The one number we do not know is the sticker price, so the natural first move is Tool #4: call it S and write both store prices in terms of S. "15% off" means you still pay 85%, so store A costs 0.85S - 90; "25% off" means you pay 75%, so store B costs 0.75S. Tool #13 then turns the sentence "store A saves 15" into a single equation, and Tool #8 keeps every term in dollars so the flat90 rebate and the percent-based discounts can be added and subtracted safely.
Name the sticker price
Naming the sticker price writes both stores with one letter.
The one thing we are asked for is the unknown, so give it a letter and let the rest of the problem describe it.
6.EE.B.6Introduce A VariableTurn percents into prices
A percent off is a multiplication, and the rebate a plain subtraction.
"Percent off" is easier as "percent you still pay": 100% - 15% = 85%, so multiply by 0.85, not 0.15.
Percent off is easier read as percent still paid, so the price is multiplied by what is left, not by the discount.
▸ Why?
A percent is a count out of a hundred, so what is taken off and what is paid add up to a hundred.
▸ Why?
The discount and the payment together make the sticker price, so naming one names the other.
Write the savings equation
The stated saving becomes one equation.
"Saves $15 at the cheaper store" is just: dearer price minus cheaper price equals 15.
7.EE.B.4Convert To AlgebraSolve for the sticker price
Solving gives 750, choice (A).
Once the two prices are lined up, the percent parts collapse into a single -0.10S and the $90 carries the equation home.
7.EE.B.4Introduce A VariableTurn "percent off" into "percent you pay," name the unknown price, and one clean equation from the savings clue hands you the sticker price.
- Name the sticker price
- Turn percents into prices
- Write the savings equation
- Solve for the sticker price