2013 AMC 12A Problem 2

Question

A softball team played ten games, scoring

1,

2,

3,

4,

5,

6,

7,

8,

9,

and

10

runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

35

40

45

50

55

Answer: C

Concepts:parity casework

Difficulty rating: 1010

Small hint:

Scoring twice the opponent is only possible when the team’s own score is even

Big hint:

The five one-run losses are the odd-scored games; each opponent scored one more

Solution:

The team can only score twice as many runs as its opponent when its own score is even. Those games have scores

2, 4, 6, 8, 10,

so their opponents scored

1 + 2 + 3 + 4 + 5 = 15.

The other five games had scores

1, 3, 5, 7, 9

and were one-run losses, so their opponents scored

2 + 4 + 6 + 8 + 10 = 30.

The total is

15 + 30 = 45.

Thus, the correct answer is C.