Question

In a tennis tournament, each person plays every other person once. In this tournament, there are twice as many right-handed players as left-handed players, but left-handed players won $40\%$ more games than right-handed players. How many total games were played?

$15$

$36$

$45$

$48$

$66$

Solution:

Say there are $L$ left-handers and $2L$ right-handers, so $3L$ players and $\binom{3L}{2}$ games. Every game has one winner, and left wins are $1.4$ times right wins, so the wins split $7 : 5$ and the total must be a multiple of $12.$ Try $L = 3:$ that's $\binom{9}{2} = 36 = 12 \cdot 3,$ with left winning $21$ and right winning $15,$ and indeed $21 = 1.4 \cdot 15.$ It's achievable. So $36$ games were played. Therefore, the answer is B.

2023 AMC 10A Problem 16

Solution:

Problem 16 in Other Years