'A' can run 1 km in 5 minutes 20 seconds and 'B' can run the same distance in 6 minutes. How many meters start can 'A' give 'B' in a kilometer race so that they finish the race together?

111.1 meter

The correct answer is Option (3) → 111.1 meter **

Speed of A:

$5$ minutes $20$ seconds $= 320$ seconds

$\text{Speed}_A=\frac{1000}{320}=\frac{25}{8}$ m/s

Speed of B:

$6$ minutes $= 360$ seconds

$\text{Speed}_B=\frac{1000}{360}=\frac{25}{9}$ m/s

Let the head start given to B be $d$ meters. Then A runs $1000$ m while B runs $(1000-d)$ m in the same time.

Time equality:

$\frac{1000}{\frac{25}{8}}=\frac{1000-d}{\frac{25}{9}}$

Simplify:

$1000 \cdot \frac{8}{25} = (1000-d)\cdot \frac{9}{25}$

$320 = \frac{9(1000-d)}{25}$

$8000 = 9(1000-d)$

$8000 = 9000 - 9d$

$9d = 1000$

$d = \frac{1000}{9} \approx 111.11$

A can give B a start of $\frac{1000}{9}$ meters (≈ 111.11 m).