At a game of billiards, A can give B 20 points in a game of 60, and A can give C 15 points in a game of 60. How many points can C give B in a game of 90?

Introduction / Context:
This billiards problem is a standard relative speed and advantage question disguised in scoring terms. A can give B a certain number of points in a fixed game and also give C a different number of points in the same sized game. From these advantages, we infer the relative strengths of A, B and C. Then, for a game of a different size, we compute how many points C can give B.

Given Data / Assumptions:

• A can give B 20 points in 60, meaning when A scores 60, B scores 40.
• A can give C 15 points in 60, meaning when A scores 60, C scores 45.
• Scores are proportional to players' skill levels or scoring rates.
• We must find how many points C can give B in a game up to 90 points.

Concept / Approach:
We treat each player's scoring ability as a rate and use ratios. From the statements, we deduce A : B and A : C. Then we derive B : C and use it to determine the relative scores when C reaches 90 points. The difference between C's and B's scores at that time is the number of points C can give B in a 90 point game.

Step-by-Step Solution:
A gives B 20 points in 60, so when A scores 60, B scores 40. Thus A : B = 60 : 40 = 3 : 2. A gives C 15 points in 60, so when A scores 60, C scores 45. Thus A : C = 60 : 45 = 4 : 3. From A : B = 3 : 2, take A = 3r and B = 2r. From A : C = 4 : 3, take A = 4s and C = 3s. Equate A: 3r = 4s, so r = 4s / 3. Then B = 2r = 2 * (4s / 3) = 8s / 3. C = 3s. Thus B : C = (8s / 3) : 3s. Multiply both sides by 3 to clear denominator: 8s : 9s = 8 : 9. So B : C = 8 : 9. In a game of 90 points, if C reaches 90, B's score is (8 / 9) * 90 = 80. Therefore, C can give B 90 − 80 = 10 points in a game of 90.

Verification / Alternative check:
We can also choose convenient scores. Suppose the skill units are set so that C scores 9 units while B scores 8 units in the same time. If C aims for 90 points, B will reach 80 points under the same conditions, confirming the 10 point difference. This is consistent with the derived ratio B : C = 8 : 9 and supports the conclusion.

Why Other Options Are Wrong:
Differences of 9, 11, 8 or 12 points do not match the 8 : 9 ratio when scaled to a 90 point target for C. For example, if the difference were 9 points, B would have to score 81 while C scores 90, which does not maintain the B : C ratio of 8 : 9. Only a 10 point difference preserves the correct ratio at that game length.

Common Pitfalls:
A common mistake is to average the advantages or directly subtract 20 and 15 without converting them into ratios. Another is to forget that the same time frame or scoring opportunity is implied when comparing players. Always convert each advantage statement into a ratio, find relative strengths and then apply them to the new game length.

Final Answer:
In a game of 90 points, C can give B 10 points.