AMC 10 · 2006 · #3
Grade 7 arithmeticA football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
Pick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Two teams scored $34$ points in total. The winning team scored $14$ points more than the losing team. Find how many points the losing team (the Panthers) scored.
Givens: The Cougars and the Panthers together scored $34$ points; The Cougars won by a margin of $14$ points, so the Cougars scored $14$ more than the Panthers; Answer choices: (A) $10$, (B) $14$, (C) $17$, (D) $20$, (E) $24$
Unknowns: The number of points the Panthers scored
Understand
Restated: Two teams scored $34$ points in total. The winning team scored $14$ points more than the losing team. Find how many points the losing team (the Panthers) scored.
Givens: The Cougars and the Panthers together scored $34$ points; The Cougars won by a margin of $14$ points, so the Cougars scored $14$ more than the Panthers; Answer choices: (A) $10$, (B) $14$, (C) $17$, (D) $20$, (E) $24$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #3 Eliminate Possibilities
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.
Execute — Answer: A
6.EE.B.6 Step 1 Name the unknown score
- Let $p$ be the number of points the Panthers scored.
- The Cougars scored $14$ more than the Panthers, so the Cougars' score is $p+14$.
- Now both scores are written in terms of one letter.
💡 Naming the smaller score once lets the bigger score be written from it, so only one unknown is left.
7.EE.B.4 Step 2 Turn the total into an equation
- The two scores add to $34$.
- Replace them with their expressions: $p+(p+14)=34$.
- Combine the two $p$ terms to get $2p+14=34$.
💡 Adding the two team scores must reproduce the given total, which pins down $p$.
7.EE.B.4 Step 3 Solve for the Panthers' score
- Undo the equation step by step.
- Subtract $14$ from both sides: $2p=20$.
- Then divide both sides by $2$: $p=10$.
- So the Panthers scored $10$ points.
💡 Peel off the $+14$ and the $\times2$ in reverse to leave $p$ alone.
6.EE.B.5 Step 4 Check both facts and pick the answer
- If the Panthers scored $10$, the Cougars scored $10+14=24$.
- The total is $10+24=34$ (correct) and the margin is $24-10=14$ (correct).
- The question asks for the Panthers, the losing team, so the answer is $10$, choice (A).
- Note $24$ is the Cougars' score and $17$ is what each would score in a tie — both are traps.
💡 A correct pair of scores must satisfy both the total and the difference at the same time.
6.EE.B.6 Let $p$ be the number of points the Panthers scored. The Cougars scored $14$ mor 7.EE.B.4 The two scores add to $34$. Replace them with their expressions: $p+(p+14)=34$. 7.EE.B.4 Undo the equation step by step. Subtract $14$ from both sides: $2p=20$. Then div 6.EE.B.5 If the Panthers scored $10$, the Cougars scored $10+14=24$. The total is $10+24= Review
Reasonableness: The Panthers are the losing team, so their score must be below half of $34$, which is $17$; the value $10$ is indeed under $17$, while choices (C) $17$, (D) $20$, and (E) $24$ are all at or above the tie line and cannot belong to the loser. Adding the found scores gives $10+24=34$ and their gap is $14$, matching both stated facts exactly.
Alternative: Reason without algebra: strip the $14$-point margin off the total first, $34-14=20$, which is the score the two teams would share if they were tied. Split it evenly, $20\div2=10$, giving the Panthers $10$; hand the $14$ back to the Cougars for $10+14=24$. Same answer (A) in two arithmetic steps.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions when solving a problem (Letting $p$ stand for the Panthers' score and writing the Cougars' score as $p+14$.)7.EE.B.4Use variables to represent quantities and construct simple equations to solve problems (Building $2p+14=34$ from the total and solving it to get $p=10$.)6.EE.B.5Understand solving an equation as finding the values that make it true; use substitution to check (Substituting $p=10$ back to confirm the total is $34$ and the margin is $14$.)
⭐ When 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.
⭐ When 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.
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