Ajay's Distance Calculation

This problem involves calculating the total distance traveled by Ajay, considering a change in his walking speed halfway through his journey.

Understanding the Journey

• Initial Speed ($v_1$): 4 km/hr
• Total Time ($T$): 12 hours
• Condition: Speed doubles after reaching exactly half the total distance.
• Goal: Find the total distance ($D$).

Calculating Time for Each Half

Let the total distance be $D$. The journey is divided into two halves based on distance:

• First half distance: $\frac{D}{2}$
• Second half distance: $\frac{D}{2}$

Let $t_1$ be the time taken for the first half and $t_2$ be the time taken for the second half.

• Speed for the first half ($v_1$): 4 km/hr
• Speed for the second half ($v_2$): $2 \times v_1 = 2 \times 4 = 8$ km/hr

Using the formula Time = Distance / Speed:

• Time for the first half ($t_1$): $\frac{D/2}{v_1} = \frac{D/2}{4} = \frac{D}{8}$ hours
• Time for the second half ($t_2$): $\frac{D/2}{v_2} = \frac{D/2}{8} = \frac{D}{16}$ hours

Solving for Total Distance

The total time is the sum of the times for both halves:

$ T = t_1 + t_2 $

Substitute the known values and expressions:

$ 12 = \frac{D}{8} + \frac{D}{16} $

To solve for $D$, find a common denominator (16):

$ 12 = \frac{2D}{16} + \frac{D}{16} $

$ 12 = \frac{3D}{16} $

Now, isolate $D$:

$ 3D = 12 \times 16 $

$ 3D = 192 $

$ D = \frac{192}{3} $

$ D = 64 $

The total distance travelled by Ajay is 64 km.