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

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.

2013 AMC 12A Problem 2

Answer: C

Difficulty rating: 1010

Solution:

Problem 2 in Other Years