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

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Copyright © 1966-2016 Alexander Bogomolny

"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

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Copyright © 1966-2016 Alexander Bogomolny

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