In a game played to 100 points, player A can give B a start of 20 points and can give C a start of 28 points. How many points can B give C in a game of 100 points?

Introduction / Context:
This is a classic problem on comparative performance in games. When we say that one player can give another a certain number of points in a fixed target game, it means that the stronger player can still reach the target while the weaker player is lagging. The question asks us to deduce the relative strengths of three players and then find the handicap one of them can give to another.

Given Data / Assumptions:
- Game target is 100 points. - A can give B 20 points, meaning when A scores 100, B scores only 80. - A can give C 28 points, meaning when A scores 100, C scores only 72. - All players score points at constant relative rates in any game length. - We must find how many points B can give C in a 100 point game.

Concept / Approach:
The idea is similar to relative speed. Here, scoring rate replaces speed. When A scores 100 while B scores 80, their scoring rates are in the ratio 100 : 80. Likewise, A and C have a rate ratio 100 : 72. Once we know these, we can find the rate ratio of B to C and then determine the handicap B can offer to C in a race to 100 points.

Step-by-Step Solution:
Step 1: From A vs B, scoring rate ratio vA : vB = 100 : 80 = 5 : 4. Step 2: From A vs C, scoring rate ratio vA : vC = 100 : 72 = 25 : 18. Step 3: Express vB and vC in terms of vA. From step 1, vB = (4/5) * vA. From step 2, vC = (18/25) * vA. Step 4: Find vB : vC = (4/5) * vA : (18/25) * vA = (4/5) : (18/25) = (4/5) * (25/18) = 100 / 90 = 10 / 9. Step 5: Let B and C play to 100 points. If B scores 100 points, C will score (9/10) * 100 = 90 points in the same time. Step 6: Therefore B can give C 10 points in a game of 100 points.

Verification / Alternative check:
We can pick a common unit model. Assume vA = 50 units. Then vB = 40 units and vC = 36 units. Time for B to reach 100 points is 100 / 40 = 2.5 units. In 2.5 units, C scores 36 * 2.5 = 90 points. So if B starts from 0 and C starts from 10, both reach 100 together. This confirms that the correct handicap from B to C is 10 points.

Why Other Options Are Wrong:
- 8 points and 14 points: These values correspond to incorrect manipulation of the rate ratios and do not satisfy the exact proportional relation. - 40 points: This is far too large and would imply C is very weak compared to B, which is not supported by the data.

Common Pitfalls:
Learners sometimes treat the given points as direct differences and try to subtract or add 20 and 28 without converting them into rate ratios. Others forget that the game length remains 100 points and mistakenly change the target. Always interpret “A can give B k points in 100” as meaning when A reaches 100, B reaches only 100 minus k, then treat points scored per unit time as proportional to these numbers.

Final Answer:
In a 100 point game, B can give C a start of 10 points.