# Billiard Handicap

Here is a word problem that involves ratios and good will:

"Do you play billiard? Care to have a game?" asked Huntington of the new member of the Town Club.

"Yes, I play," replied McClintock, "But I am rather a duffer. My friend Chadwick gives me 25 points in 100, and then we play about even."

"Well, I am perfectly willing to give you a proper handicap. I give Chadwick 20 points in 100. Now let's see how many points should I give you?"

What is the correct answer, assuming that the stated handicaps are fare.

Solution

"Do you play billiard? Care to have a game?" asked Huntington of the new member of the Town Club.

"Yes, I play," replied McClintock, "But I am rather a duffer. My friend Chadwick gives me 25 points in 100, and then we play about even."

"Well, I am perfectly willing to give you a proper handicap. I give Chadwick 20 points in 100. Now let's see how many points should I give you?"

What is the correct answer, assuming that the stated handicaps are fare.

### Solution

If Huntington gives Chadwick 20 points in 100, then Chadwick is expected to make 80 points while Huntington makes 100, thus scoring at 4/5 of Huntington's rate. Similarly, McClintock scores at 3/4,3/4,4/5,3/5 the rate of Chadwick. Then, McClintock should score 4/5×3/4=3/5,3/4,4/5,3/5 the rate of Huntington. Thus, McClintock gains 60,60,50,40,30 points to Huntington's 100, such that Huntington should give McClintock 40,60,50,40,30 points in 100.

### References

1. J. G. McLoughlin et al, Jim Totten's Problems of the Week, World Scientific, 2013, #72