AMC 10 · 2006 · #3
Grade 7 algebraPick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Let p be the Panthers' score. The Cougars scored 14 more, so their score is p+14 — both scores now use one letter.
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 two scores add to 34, so p+(p+14)=34, which combines to 2p+14=34.
Adding the two team scores must reproduce the given total, which pins down p.
Adding the two team scores must reproduce the given total, which pins down the unknown.
▸ Why?
The two scores together make up the whole total and nothing else does.
▸ Why?
Peeling the same amount off both sides keeps the equation true, so the unknown can be freed.
Solve for the Panthers' score
Subtract 14 from both sides to get 2p=20, then divide by 2: p=10.
Peel off the +14 and the ×2 in reverse to leave p alone.
7.EE.B.4Introduce A VariableCheck both facts and pick the answer
Then the Cougars have 24: the total 10+24=34 and the margin 24-10=14 both check, so the loser's 10 is (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