Three persons, A, B and C, took part in a 2000 m race. To reach the finish line at the same time, A can give to B a start of 500 m and to C a start of 300 m. How many metres of start can C give to B in a 3400 m race?

The correct option is 400.

Let us break down the problem step-by-step to understand why this is the correct answer.

First, we analyze the relationship between the distances covered by the runners in the same duration of time.

We are given a race of 2000 m. For A, B, and C to reach the finish line at the same time:
A can give B a start of 500 m. This means when A runs the full distance of 2000 m, B only needs to run:
$2000-500=1500m$
Similarly, A can give C a start of 300 m. This means when A runs the full distance of 2000 m, C only needs to run:
$2000-300=1700m$

Since they all reach the finish line at the same time, the ratio of the speeds of A, B, and C is equal to the ratio of the distances they cover in the same time interval.
Let $D_A$, $D_B$, and $D_C$ be the distances covered by A, B, and C respectively in the same time duration.
We have:
$D_A=2000m$
$D_B=1500m$
$D_C=1700m$

From this, we can find the ratio of the distance covered by B to the distance covered by C:
$\frac{D_B}{D_C}=\frac{1500}{1700}=\frac{15}{17}$

This ratio tells us that when C covers 17 units of distance, B covers 15 units of distance in the same time.

Now, we want to find out how many metres of start C can give B in a 3400 m race so that they finish at the same time.
In this case, C must run the full length of the race, which is 3400 m. Let the distance B runs in the same time be $X_B$.
Using the constant ratio of their speeds, we can set up the proportion:
$\frac{X_B}{3400}=\frac{15}{17}$

Solving for $X_B$:
$X_B=3400\times \frac{15}{17}$
$X_B=200\times 15$
$X_B=3000m$

Thus, while C runs the full 3400 m, B runs 3000 m.
The head start that C can give to B is the difference between the total distance C runs and the distance B runs:
$\text{Start given to B}=3400-3000=400m$

Therefore, C can give B a start of 400 m in a 3400 m race.