A runs 25\% faster than B and is able to allow B a lead of 7 m to end a race in dead heat. What is the length of the race?

Question

Accepted Answer

**Solution:**

1. Let the length of the race be *d* meters.

2. Let B's speed be *x* m/s.

3. Since A runs 25\% faster than B, A's speed = $ x \times \frac{125}{100} = \frac{5x}{4} $.

4. B is given a lead of 7 m, so B starts 7 m ahead.

5. Both A and B finish the race at the same time (dead heat), so their times are equal.

6. Time taken by A to run *d* meters:
   $$
   t_A = \frac{d}{\frac{5x}{4}} = \frac{4d}{5x}
   $$

7. Time taken by B to run $ d - 7 $ meters:
   $$
   t_B = \frac{d - 7}{x}
   $$

8. Since $ t_A = t_B $, we have:
   $$
   \frac{4d}{5x} = \frac{d - 7}{x}
   $$

9. Multiply both sides by $ 5x $:
   $$
   4d = 5(d - 7)
   $$

10. Expand and simplify:
    $$
    4d = 5d - 35 \ 
    35 = 5d - 4d = d
    $$

**Answer:** The length of the race is $ \boxed{35 \text{ m}} $.

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