Discussion
At a game of billiards, A can give B 12 points in a game of 40 and A can give C 10 points in a game of 50. How many points can C give B in a game of 80?
Answer: A
In these problems the two players are compared by the scores they reach in the same time, so their rates are in the ratio of those scores.
The handicaps are over different totals, so scale both against A.
\[ A\,40 : B\,28 \ \Rightarrow\ B = \tfrac{7}{10}A \]
\[ A\,50 : C\,40 \ \Rightarrow\ C = \tfrac{8}{10}A \]
\[ B : C = 7 : 8 \]
C is the stronger, so in a game of 80, when C reaches 80:
\[ B = 80 \times \frac{7}{8} = 70 \]
\[ \text{C can give B} = 80 - 70 = 10 \text{ points} \]
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