2006 AMC 10B Problem 3

P1-10~1 minPrint

Linear Equations & Word Problems Systems of Equations Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Take away the 14-point winning margin from the total 34. What is left, 20, is split equally between the two teams, so the losing team has 10.

Solution

The Cougars won, so they scored $14$ more points than the Panthers.
Picture taking those extra $14$ points away from the Cougars. Then both teams have the same score, and the total drops to $34 - 14 = 20$ .
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.
Check: the Cougars scored $10 + 14 = 24$ . Then $24 + 10 = 34$ and $24 - 10 = 14$ . Both facts match.

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

Why this works

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.

Alternative approach

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 $p = 34 - 24 = 10$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 5: Multi-Variable Linear Equations (§5.2, §5.3, §5.6)
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Intermediate Algebra, Chapter 1: Basic Techniques for Solving Equations (§1.4)