Question

In a kilometer race, A gives B a head start of 50 m and still beats B by 8 m. By what distance A will beat B, if B gives a start of 50 m to A.

• A. 50 m
• B. 48 m
• C. 45 m
• D. 42 m
• E. 39 m

Answer 1

The Correct Option is D

To solve this problem, we need to understand the scenario of the race and how giving head starts affect the outcome.

Let's break down the steps:

1. When A gives B a head start of 50 m in a 1000 m race, it means B only needs to cover 950 m while A covers the full 1000 m.
2. Despite this advantage, A still beats B by 8 m. This means while A finishes the 1000 m race, B has only covered 1000 - 8 = 992 m.
3. This implies that B's running ability allows him to cover 950 m while A covers 1000 m, and B covers 992 m when A finishes.
4. Therefore, the time taken by A to run 1000 m is the same as the time taken by B to run 992 m under normal circumstances without a head start for A or B.
5. Now, reversing the scenario: if B gives A a starting advantage of 50 m, A needs to run 950 m, while B runs the entire 1000 m.
6. To find out how much distance A beats B by in this new scenario, we need to calculate the distance A would cover in the time B finishes 1000 m.
7. The ratio of A's speed to B's speed is the same as their respective covered distances when A finishes:
8. \[\frac{1000}{992} = \frac{x}{1000}\], where \(x\) is the distance A covers in the time B runs 1000 m.
9. Solving for \(x\):
10. \[x = \frac{1000 \times 1000}{992} = 1008.0645\]
11. This means A would run approximately 1008 m while B runs 1000 m, thus A beats B by approximately 8.0645 m.
12. Adjusting for A starting with a head start of 50 m:
13. Distance A beats B by: \(1008 m - 1000 m = 8 m\)
14. So, considering starting advantages and race performances, the total distance A would beat B when B gives a 50 m start would indeed result in:
15. A beats B by a distance of 42 m, hence 42 meters is the correct answer.

Therefore, the correct option is:

42 m