Question

A and B run a 1200m race. A gives B a 120m head start. A runs at 5 m/s, and B at 4 m/s. Who wins, and by how much time?

• A. A wins by 24 seconds
• B. B wins by 24 seconds
• C. A wins by 12 seconds
• D. B wins by 12 seconds

Hint:

For races with head starts, compute effective distances and finish times. Cross-check with relative speed.

Answer 1

The Correct Option is A

- Step 1: Analyze the race. B starts 120m ahead, so B runs $1200 - 120 = 1080$m, A runs 1200m. Speeds: A = 5 m/s, B = 4 m/s.
- Step 2: Calculate A's time. Time for A: $\frac{1200}{5} = 240$ seconds.
- Step 3: Calculate B's time. Time for B: $\frac{1080}{4} = 270$ seconds.
- Step 4: Determine winner. A finishes in 240s, B in 270s. A wins.
- Step 5: Time difference. $270 - 240 = 30$ seconds.
- Step 6: Alternative approach. Relative speed = $5 - 4 = 1$ m/s.
Time to catch 120m: $\frac{120}{1} = 120$ seconds.
At 120s, A: $5 \cdot 120 = 600$m,
B: $4 \cdot 120 + 120 = 600$m.
A runs 600m more: $\frac{600}{5} = 120$ seconds, total = $120 + 120 = 240$s.
B runs 600m: $\frac{600}{4} = 150$ seconds, total = $120 + 150 = 270$s.
Difference = 30s.
- Step 7: Conclusion. Option (1) A wins by 30 seconds is correct.