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: 1070

Solution:

Note that if they scored twice as many runs as their opponents, then they scored an even number of runs.

This means in the games where they scored $2, 4, 6, 8,$ and $10$ runs, their opponents scored $1, 2, 3, 4,$ and $5$ runs respectively.

This sums to $1 + 2 + 3 + 4 + 5 = 15.$

In the other games, their opponents scored $2, 4, 6, 8,$ and $10$ runs.

This sums to $2 + 4 + 6 + 8 + 10 = 30.$

The total number of runs is then $15 + 30 = 45.$ Thus, C is the correct answer.

2013 AMC 10A Problem 4

Answer: C

Difficulty rating: 1070

Solution:

Problem 4 in Other Years