Problem Setup: Budhan's Journey

We need to find Budhan's walking speed. Let '$S_w$' represent the walking speed in km/h and '$S_c$' represent the cycling speed in km/h.

The total time for both days is 2 hours.

Day 1 Calculation

Budhan cycles for $\frac{1}{4}$ of the time and walks for the rest.

• Time cycling = $\frac{1}{4} \times 2$ hours = 0.5 hours.
• Time walking = $2 - 0.5$ hours = 1.5 hours.
• Total distance = 19 km.
• Using Distance = Speed $\times$ Time, we get the equation: $ S_c \times 0.5 + S_w \times 1.5 = 19 \quad \cdots (1) $

Day 2 Calculation

Budhan cycles for $\frac{1}{2}$ of the time and walks for the rest.

• Time cycling = $\frac{1}{2} \times 2$ hours = 1 hour.
• Time walking = $2 - 1$ hour = 1 hour.
• Total distance = 26 km.
• The equation for Day 2 is: $ S_c \times 1 + S_w \times 1 = 26 \quad \cdots (2) $

Solving for Walking Speed ($S_w$)

We have a system of two linear equations:

1. $0.5 S_c + 1.5 S_w = 19$
2. $S_c + S_w = 26$

From Equation (2), we can express the cycling speed '$S_c$' in terms of walking speed '$S_w$': $ S_c = 26 - S_w $

Now, substitute this expression for '$S_c$' into Equation (1): $ 0.5 (26 - S_w) + 1.5 S_w = 19 $

Distribute the 0.5:

$ 13 - 0.5 S_w + 1.5 S_w = 19 $

Combine the '$S_w$' terms:

$ 13 + 1.0 S_w = 19 $

Isolate the '$S_w$' term:

$ S_w = 19 - 13 $

$ S_w = 6 $

The walking speed is 6 km/h.