To find the average speed for the whole journey, you can use the formula for average speed:

$\text{AverageSpeed}=\frac{\text{TotalDistance}}{\text{TotalTime}}$

Let’s assume the one-way distance between Nagpur and Allahabad is $D$. The total distance for the round trip is $2D$, as the train is traveling from Nagpur to Allahabad and then back from Allahabad to Nagpur.

Let ${t}_{1}$ be the time taken to travel from Nagpur to Allahabad, and ${t}_{2}$ be the time taken to travel back from Allahabad to Nagpur.

The average speed is given by:

$\text{AverageSpeed}=\frac{2D}{{t}_{1}+{t}_{2}}$

To find ${t}_{1}$ and ${t}_{2}$, you can use the formula $\text{time}=\frac{\text{distance}}{\text{speed}}$

For the journey from Nagpur to Allahabad: ${t}_{1}=\frac{D}{100}$

For the journey back from Allahabad to Nagpur: ${t}_{2}=\frac{D}{150}$

Now, substitute these values into the average speed formula:

$\text{AverageSpeed}=\frac{2D}{\frac{D}{100}+\frac{D}{150}}$

To simplify, you can find a common denominator and then cancel out terms:

$\text{AverageSpeed}=\frac{2D}{\frac{3D+2D}{300}}$

$\text{AverageSpeed}=\frac{2D}{\frac{5D}{300}}$

$\text{AverageSpeed}=\frac{2D\times 300}{5D}$

$\text{AverageSpeed}=\frac{600}{5}$

$\text{AverageSpeed}=120\text{\hspace{0.17em}}\text{km/h}$

So, the average speed during the whole journey is 120 km/h.

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