AMC 10 · 2006 · #3
Grade 7 algebraPick an answer.
There are two unknown scores tied together by two facts — a total and a difference — so Tool #4 (Introduce a Variable) names the Panthers' score p, writes the Cougars' score as p+14, and turns the words into a single equation. Tool #3 (Eliminate Possibilities) then checks the found number against both facts, which also guards the common trap of reporting the winner's score 24 or the tie value 17 instead of the loser's score.
Name the unknown score
Naming the smaller score writes the other as a plain addition.
Naming the smaller score once lets the bigger score be written from it, so only one unknown is left.
6.EE.B.6Introduce A VariableTurn the total into an equation
The total becomes one equation.
Adding the two team scores must reproduce the given total, which pins down p.
7.EE.B.4Introduce A VariableSolve for the Panthers' score
Solving gives 10.
Peel off the +14 and the ×2 in reverse to leave p alone.
Peeling off the added fourteen and then the doubling in reverse order leaves the unknown score alone.
▸ Why?
Each operation is undone by its opposite, so reversing the chain recovers the number it started from.
▸ Why?
Doing the same thing to both sides keeps the equation true while the unknown is stripped bare.
Check both facts and pick the answer
Both given facts check out, so the answer is 10, choice (A).
A correct pair of scores must satisfy both the total and the difference at the same time.
6.EE.B.5Eliminate PossibilitiesWhen you know a total and a difference, name the smaller amount with one letter, write the bigger one as that letter plus the gap, and one equation gives both.
- Name the unknown score
- Turn the total into an equation
- Solve for the Panthers' score
- Check both facts and pick the answer