A runs 3/4th faster than B.One of them was placed some metres ahead.How far should the finishing post be placed so that both of them finish at the same time?

To solve the problem of how far the finishing post should be placed so that both A and B finish at the same time, we need to analyze their speeds and the distance involved. Let's break it down step by step.

Understanding the Speed Relationship

We know that A runs 3/4th faster than B. This means if we let B's speed be represented as v, then A's speed can be expressed as:

Speed of A = v + (3/4)v = (7/4)v

Setting Up the Problem

Let's denote the distance that B has to run as d. Since one of them is placed some meters ahead, we can assume that A has a head start of h meters. Therefore, the distance A has to run is:

Distance for A = d - h

Time Taken to Finish

The time taken by both A and B to finish the race can be expressed using the formula:

Time = Distance / Speed

For B, the time taken to finish is:

Time for B = d / v

For A, the time taken is:

Time for A = (d - h) / (7/4)v

Equating the Times

Since we want both A and B to finish at the same time, we can set their times equal to each other:

d / v = (d - h) / (7/4)v

Simplifying the Equation

We can eliminate v from both sides (assuming v is not zero), leading to:

d = (d - h) * (4/7)

Now, let's multiply both sides by 7 to eliminate the fraction:

7d = 4(d - h)

Expanding the right side gives:

7d = 4d - 4h

Rearranging the Equation

Now, we can rearrange the equation to isolate h:

7d - 4d = -4h

3d = -4h

Dividing both sides by -4 gives:

h = -3d / 4

Finding the Finishing Post Distance

Now, we need to determine how far the finishing post should be placed. Since h represents the head start that A has, we can express the total distance to the finishing post for B as:

Finishing Post Distance = d + h

Substituting the value of h we found:

Finishing Post Distance = d - 3d / 4 = d / 4

Final Calculation

Thus, the finishing post should be placed at a distance of:

d / 4 meters ahead of B's starting point for both A and B to finish at the same time.

In summary, by analyzing their speeds and the distance they need to cover, we determined that the finishing post should be set at a quarter of the distance that B has to run, ensuring a fair race where both competitors finish simultaneously.