To solve this problem, we need to determine the average speeds of Raghav and Manish, then find the difference between them.

Step 1: Calculate Raghav's Average Speed

Raghav travels the same distance, D, at different speeds for each quarter. The distances for the four quarters of his journey are D/4 each. The speeds are 10 km/h, 15 km/h, 20 km/h, and 30 km/h, respectively.

The time taken for each part is calculated as:

1. First quarter: t_1 = \frac{D/4}{10} = \frac{D}{40}
2. Second quarter: t_2 = \frac{D/4}{15} = \frac{D}{60}
3. Third quarter: t_3 = \frac{D/4}{20} = \frac{D}{80}
4. Fourth quarter: t_4 = \frac{D/4}{30} = \frac{D}{120}

Total time, T_R = t_1 + t_2 + t_3 + t_4 = \frac{D}{40} + \frac{D}{60} + \frac{D}{80} + \frac{D}{120}.

To find a common denominator:

• The least common multiple of 40, 60, 80, and 120 is 240.
• Convert each term to have a denominator of 240:
  • \frac{D}{40} = \frac{6D}{240}
  • \frac{D}{60} = \frac{4D}{240}
  • \frac{D}{80} = \frac{3D}{240}
  • \frac{D}{120} = \frac{2D}{240}
• Therefore, T_R = \frac{6D + 4D + 3D + 2D}{240} = \frac{15D}{240} = \frac{D}{16}.

The average speed of Raghav, y is given by: y = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{D/16} = 16 \text{ km/h}.

Step 2: Calculate Manish's Average Speed

Manish divides his total journey time into four equal quarters and travels at different speeds: 10 km/h, 15 km/h, 20 km/h, and 30 km/h.

Let T be the total time taken by Manish. Each quarter of time is T/4.

The distances traveled in each quarter are:

1. First quarter: d_1 = 10 \times T/4 = 2.5T
2. Second quarter: d_2 = 15 \times T/4 = 3.75T
3. Third quarter: d_3 = 20 \times T/4 = 5T
4. Fourth quarter: d_4 = 30 \times T/4 = 7.5T

Total distance covered by Manish, D_M = 2.5T + 3.75T + 5T + 7.5T = 18.75T.

The average speed of Manish, z is: z = \frac{D_M}{T} = \frac{18.75T}{T} = 18.75 \text{ km/h}.

Step 3: Difference in Average Speed

We calculate the difference between the averages of Manish and Raghav:

z - y = 18.75 - 16 = 2.75.

Thus, the value of (z-y) is 2.75.