Step 1: Identify distance and speeds for each stretch

The total distance is divided into four equal stretches of $d = 3\text{ km}$ each. The respective speeds are $v_1 = 10\text{ km/h}$, $v_2 = 20\text{ km/h}$, $v_3 = 30\text{ km/h}$, and $v_4 = 60\text{ km/h}$.

$d = 3\text{ km}, \quad v_1 = 10, v_2 = 20, v_3 = 30, v_4 = 60$

Step 2: Calculate the time taken for each stretch

Time taken for each stretch is given by $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$. Thus, $t_1 = \frac{3}{10}$, $t_2 = \frac{3}{20}$, $t_3 = \frac{3}{30} = \frac{1}{10}$, and $t_4 = \frac{3}{60} = \frac{1}{20}$.

$t_1 = \frac{3}{10}, \quad t_2 = \frac{3}{20}, \quad t_3 = \frac{3}{30}, \quad t_4 = \frac{3}{60}$

Step 3: Find total distance and total time

Total distance = $3 + 3 + 3 + 3 = 12\text{ km}$. Total time = $\frac{3}{10} + \frac{3}{20} + \frac{3}{30} + \frac{3}{60} = \frac{18 + 9 + 6 + 3}{60} = \frac{36}{60} = \frac{3}{5}\text{ hours}$.

$\text{Total Distance} = 12\text{ km}, \quad \text{Total Time} = \frac{36}{60} = \frac{3}{5}\text{ hours}$

Step 4: Calculate average speed

Applying the average speed formula: $v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{12}{3/5} = 12 \times \frac{5}{3} = 20\text{ km/h}$.

$v_{avg} = \frac{12}{\frac{3}{5}} = 20\text{ km/h}$