A car travels $\frac{3}{4}$th of the distance at a speed of $60 \text{

Part 2: Distance $D_2 = D – D_1 = D – \frac{3}{4}D = \frac{1}{4}D$. Speed $S_2 = v \text{ km/hr}$.
Time taken for Part 2: $T_2 = \frac{D_2}{S_2} = \frac{\frac{1}{4}D}{v} = \frac{D}{4v}$ hours.

Total distance for the journey is $D$.
Total time for the journey is $T_{total} = T_1 + T_2 = \frac{D}{80} + \frac{D}{4v}$.

The average speed for the full journey is given as $50 \text{ km/hr}$.
Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$
$50 = \frac{D}{\frac{D}{80} + \frac{D}{4v}}$

We can factor out $D$ from the denominator:
$50 = \frac{D}{D\left(\frac{1}{80} + \frac{1}{4v}\right)}$
$50 = \frac{1}{\frac{1}{80} + \frac{1}{4v}}$

Taking the reciprocal of both sides:
$\frac{1}{50} = \frac{1}{80} + \frac{1}{4v}$

Now, solve for $v$:
$\frac{1}{4v} = \frac{1}{50} – \frac{1}{80}$

Find a common denominator for the right side, which is 400:
$\frac{1}{4v} = \frac{8}{400} – \frac{5}{400}$
$\frac{1}{4v} = \frac{8 – 5}{400}$
$\frac{1}{4v} = \frac{3}{400}$

Cross-multiply:
$4v \times 3 = 1 \times 400$
$12v = 400$

Divide by 12:
$v = \frac{400}{12}$

Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 4:
$v = \frac{400 \div 4}{12 \div 4} = \frac{100}{3}$

The value of $v$ is $\frac{100}{3} \text{ km/hr}$.