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

Introduction / Context:
This question is similar to the earlier point game problems but now includes a different target score. A can give handicaps to two other players in a 60 point game, and we must find the handicap one of those players can give to the other in a 90 point game. It checks the ability to use ratios of scoring rates and to scale them to a different game length.

Given Data / Assumptions:
- In a game to 60 points, A can give B 15 points. - In the same game type, A can give C 20 points. - This means that when A scores 60 points, B has scored only 45 and C has scored only 40. - Scores are proportional to scoring rates, which are constant for each player. - We must find how many points B can give C in a 90 point game.

Concept / Approach:
The ratio of scoring rates is derived from the scores at the moment A reaches 60 points. A to B is 60 : 45, and A to C is 60 : 40. We then compute B : C and use that ratio in a 90 point game where B plays to 90 points. By seeing how many points C scores in that time, we can work out the handicap B can offer to C so that both finish together at 90 points.

Step-by-Step Solution:
Step 1: From A and B, vA : vB = 60 : 45 = 4 : 3. Step 2: From A and C, vA : vC = 60 : 40 = 3 : 2. Step 3: Write vB = (3/4) * vA and vC = (2/3) * vA. Step 4: Ratio vB : vC = (3/4) : (2/3) = (3/4) * (3/2) = 9 : 8. Step 5: Thus vB = 9k and vC = 8k for some constant k. Step 6: In a 90 point game, time for B to score 90 points is TB = 90 / (9k) = 10 / k. Step 7: In this time, C scores points = vC * TB = 8k * (10 / k) = 80 points. Step 8: Therefore B can give C a start of 90 - 80 = 10 points in a 90 point game.

Verification / Alternative check:
Assume vB = 9 and vC = 8. When B scores 90, time taken is 90 / 9 = 10 units. In 10 units C scores 8 * 10 = 80. If C starts at 10 points, C also reaches 90 at the same time. This shows that a 10 point handicap from B to C in a 90 point game is exactly correct.

Why Other Options Are Wrong:
- 30 points and 20 points: These are too large and would give C an unrealistic advantage, causing C to finish well before B. - 12 points: This does not match the 9 : 8 scoring rate ratio and would lead to mismatched final scores.

Common Pitfalls:
A common error is to scale the handicaps directly from 60 to 90 by simple proportion, for example taking 15 or 20 and scaling them by 1.5. This approach ignores the one step where we must first convert the handicaps into scoring rate ratios. Another mistake is to forget that the target score has changed and still work with 60 instead of 90 when computing final points.

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