Problem Analysis

To solve this problem, we need to find the total length of the racecourse. The key condition is that both runners, A and B, finish the race in the exact same amount of time.

When the time taken is equal for two runners, the ratio of the distances they cover is equal to the ratio of their speeds.
Distance_A / Distance_B = Speed_A / Speed_B

1. Determine the Ratio of Speeds

We are given that A runs $1\frac{2}{3}$ times as fast as B.

$1\frac{2}{3} = \frac{5}{3}$

This means Speed_A = $(\frac{5}{3}) \times$ Speed_B. The ratio of their speeds is:

Speed_A / Speed_B = 5 / 3

2. Define the Distances

Let 'D' be the total distance to the winning post (in meters).

• A must run the full distance: Distance_A = D
• B gets a head start of 80 m, so B has to run a shorter distance: Distance_B = D - 80

3. Set up and Solve the Equation

Using the principle that the ratio of distances equals the ratio of speeds, we can set up the following equation:

D / (D - 80) = 5 / 3

Now, we cross-multiply to solve for D:

3 × D = 5 × (D - 80)

3D = 5D - 400

400 = 5D - 3D

400 = 2D

D = 200

Conclusion

The winning post must be 200 m away for A and B to reach it at the same time.