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?

Correct Option (1)

The total race distance is 500 metres.

Runner A covers the full race distance, as A wins. Therefore, distance covered by A = 500 m.

Runner B starts 45 metres ahead of A. When A finishes the 500-metre race, B is 35 metres behind A. This implies that B's position is 35 metres short of the finish line from A's perspective. Considering B's initial head start of 45 metres, the total distance B is behind the finish line, relative to A's starting point, is the 35 metres A is ahead of B plus the 45 metres A initially started behind B's position. Thus, B's effective deficit from the finish line is 35 m + 45 m = 80 m.

Therefore, the distance covered by B = Total race distance - B's effective deficit = 500 m - 80 m = 420 m.

Since both runners start at the same time and run for the same duration, the ratio of their speeds is equivalent to the ratio of the distances they cover:

$$\frac{\text{Speed of A}}{\text{Speed of B}}=\frac{\text{Distance covered by A}}{\text{Distance covered by B}}$$

$$\frac{\text{Speed of A}}{\text{Speed of B}}=\frac{500}{420}$$

Simplifying the ratio:

$$\frac{500}{420}=\frac{50}{42}=\frac{25}{21}$$

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

Incorrect Options:

Options 2, 3, and 4 are incorrect as they do not reflect the accurate calculation of distances covered by A and B based on the given conditions. These options would result from misinterpreting the starting advantage or the final positions of the runners, leading to an incorrect ratio of distances.