At a game of billiards, A can give B 15 points in a game of 60, and A can give C 20 points in a game of 60. In a game of 90 points, how many points can B give C as a start so that they finish together?

Introduction / Context:
This is a classic billiards handicap problem involving relative scoring rates. When we say that A can give B 15 points in 60, it means that when A reaches 60 points, B has only 45. Similarly, A can give C 20 points in 60, meaning that when A reaches 60, C has only 40. Using these relationships, you are asked to find how many points B can give C in a longer game of 90 so that they still finish at the same time. The question tests understanding of proportional reasoning and relative rates.

Given Data / Assumptions:

• When A scores 60 points in a game, B has 45 points.
• When A scores 60 points, C has 40 points.
• This implies A, B, and C have different scoring rates.
• We consider a new game where the target is 90 points.
• We must find how many points B can give C as a start in this 90 point game so that they finish together.

Concept / Approach:
We interpret points given as a measure of relative scoring speed:

• Rate of scoring is directly proportional to points scored in a given time.
• When A scores 60, B scores 45, C scores 40 in the same time.
• This gives the ratio of their scoring rates: A : B : C = 60 : 45 : 40.
• Simplify to get base speed ratio, then compare B and C.
• Use these rates to find the handicap B can give C in a 90 point game.

Step-by-Step Solution:
Step 1: From given information, when A scores 60, B scores 45, C scores 40 in the same duration. Step 2: So scoring rate ratio A : B : C = 60 : 45 : 40. Step 3: Simplify, dividing by 5: A : B : C = 12 : 9 : 8. Step 4: Focus only on B and C. Their rate ratio B : C = 9 : 8. Step 5: In a game to 90 points, suppose B plays without a start and needs to reach 90 points from zero. Step 6: Let the time for B to reach 90 points be T. Step 7: Then B's rate r_B = 90 / T, and C's rate r_C = (8 / 9) * r_B since B : C = 9 : 8. Step 8: In the same time T, C will score points = r_C * T = (8 / 9) * 90 = 80 points. Step 9: So when B reaches 90 points, C would naturally reach 80 points if both started from zero at the same time. Step 10: To make them finish together in a 90 point game, B must give C a start equal to this difference: 90 - 80 = 10 points.

Verification / Alternative check:
You can verify by simulating the game:

• Let B start from 0 and C start from 10 points.
• B needs to score 90 points; C needs to go from 10 to 90, so C must score 80 points.
• Time for B to score 90 at rate r_B is T.
• At rate r_C = (8 / 9) * r_B, in time T, C scores (8 / 9) * 90 = 80 points.
• So C goes from 10 to 10 + 80 = 90 points exactly when B reaches 90.

Therefore, with a 10 point start, both B and C finish the 90 point game together.

Why Other Options Are Wrong:
30 points or 20 points would give C too large a head start, making C finish earlier than B.
12 points is slightly larger than needed and would again let C win early. The detailed rate calculations show that the exact difference in natural scores by time T is 10 points, so any larger or smaller handicap fails to produce a simultaneous finish.

Common Pitfalls:
Some learners misread "give 15 points in 60" as 15% or confuse points with percentages. Others try to subtract 15 and 20 directly or average them to find a start, which does not capture the underlying rate relationship. Always interpret such statements as "when A scores 60, B (or C) scores fewer points in the same time," then use those numbers to derive a proper rate ratio before analyzing the new game.

Final Answer:
In a 90 point game, B can give C a start of 10 points so that they finish together.