2006 AMC 10B Problem 3

1. The Cougars won, so they scored $14$ more points than the Panthers.
2. Picture taking those extra $14$ points away from the Cougars. Then both teams have the same score, and the total drops to $34 - 14 = 20$.
3. Two equal scores that add to $20$ are $10$ each. That equal score is exactly the Panthers' score, since we only took points away from the Cougars.
4. Check: the Cougars scored $10 + 14 = 24$. Then $24 + 10 = 34$ and $24 - 10 = 14$. Both facts match.

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

When you know the total of two amounts and how far apart they are, remove the difference, split what is left in half, and you have the smaller amount. Add the difference back to get the larger one. Always reread the question to see which of the two it wants.

With letters: let $c$ be the Cougars' score and $p$ the Panthers'. Then $c + p = 34$ and $c - p = 14$. Adding the two equations gives $2c = 48$, so $c = 24$, and then $p = 34 - 24 = 10$.

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 $\boxed{\textbf{(A)}\ 10}$.