In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time ?

The correct option is 25 : 21.

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

Step 1: Understand the distances covered by both runners.
The total length of the race is 500 metres. Since A wins the race, A must cover the entire distance of the race.
Therefore, the distance covered by A, denoted as _DA, is:
_DA = 500 metres

Now, let us find the distance covered by B, denoted as _DB. We are given two conditions:
1. B starts 45 metres ahead of A.
2. A wins the race while B is still 35 metres behind the finish line.

Because B starts 45 metres ahead, B only needs to cover 500 - 45 = 455 metres to reach the finish line. However, when A reaches the finish line (completing 500 metres), B is still 35 metres behind the finish line.
Thus, the actual distance covered by B during this time is:
_DB = 500 - 45 - 35 = 420 metres

Step 2: Relate distance to speed.
We are given that both A and B start at the same time. Since they start at the same time and we are comparing their positions at the exact moment A finishes, the time duration (t) for which both run is the same.

Using the formula for speed (Speed = Distance / Time), we know that:
Distance = Speed × Time

Since the time t is constant for both runners, the ratio of their distances is directly proportional to the ratio of their speeds:
$$\frac{{\mathrm{Speed\; of\; A}}_{}}{{\mathrm{Speed\; of\; B}}_{}}=\frac{D_A}{D_B}$$

Step 3: Calculate the ratio.
Substituting the distances we found into the ratio:
$$\frac{{\mathrm{Speed\; of\; A}}_{}}{{\mathrm{Speed\; of\; B}}_{}}=\frac{500}{420}$$

Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 20:
$$\frac{500}{420}=\frac{500/20}{420/20}=\frac{25}{21}$$

Therefore, the ratio of the speeds of A to B is 25 : 21.