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2006 AMC 10B Problem 3

2006 AMC 10B Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

A 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?

A)
$10$
B)
$14$
C)
$17$
D)
$20$
E)
$24$

Answer

Key insight

Sum and difference are given, so the smaller score is half of (sum minus difference): (34 - 14)/2 = 10.

Solution

Let $c$ and $p$ be the Cougars' and Panthers' scores. We know $c + p = 34$ and $c - p = 14$ .

Subtracting the second equation from the first eliminates $c$ :
$2p = 34 - 14 = 20, \qquad p = 10.$
(Then $c = 24$ , and indeed $24 + 10 = 34$ and $24 - 10 = 14$ .)

The Panthers scored $\boxed{\textbf{(A)}\ 10}$ .

Why this works

Whenever two quantities have a known sum $S$ and known difference $D$ , the larger is $\frac{S + D}{2}$ and the smaller is $\frac{S - D}{2}$ . Memorizing this pattern turns a whole class of word problems into one line of arithmetic.

The trap

Reporting the Cougars' score 24 instead of the Panthers' score, or halving the wrong combination to get 17.

Common mistakes

Reporting the Cougars' score 24 instead of the Panthers' score, or halving the wrong combination to get 17.
Treating "won by 14" as "scored 14" and answering (B).

Techniques

Set up the equation/formula and compute; no special trick needed