A person completes half of its his journey with speed v1 and rest half with speed v2 . The average speed of the person is

The correct option is v = 2v₁v₂/(v₁+v₂).

To find the average speed of the person, we use the fundamental definition of average speed:
$\mathrm{Average\; Speed}=\frac{\mathrm{Total\; Distance\; Travelled}}{\mathrm{Total\; Time\; Taken}}$

Let the total distance of the journey be $2d$. Therefore, the journey is divided into two equal halves, each of distance $d$.

For the first half of the journey:
- Distance = $d$
- Speed = $v_1$
- Time taken ($t_1$) = $\frac{d}{v_1}$

For the second half of the journey:
- Distance = $d$
- Speed = $v_2$
- Time taken ($t_2$) = $\frac{d}{v_2}$

Now, we calculate the total time taken ($T$) for the entire journey:
$T=t_1+t_2=\frac{d}{v_1}+\frac{d}{v_2}$

Taking $d$ common and finding a common denominator:
$T=d(\frac{1}{v_1}+\frac{1}{v_2})=d\left(\frac{v_1+v_2}{v_1{v}_2}\right)$

Substituting the total distance ($2d$) and total time ($T$) into the average speed formula:
$v=\frac{2d}{d\left(\frac{v_1+v_2}{v_1{v}_2}\right)}$

Canceling the common term $d$ from the numerator and denominator:
$v=\frac{2}{\frac{v_1+v_2}{v_1{v}_2}}$

Simplifying the fraction gives the final expression for the average speed:
$v=\frac{2v_1{v}_2}{v_1+v_2}$