Understanding Average Speed Calculation

The problem asks us to find the average speed for a journey where Tanmay covers three equal distances at different speeds: 2 km/h, 3 km/h, and 4 km/h. The average speed is defined as the total distance covered divided by the total time taken.

Calculating Total Time for Each Distance

Let's assume the distance for each part of the journey is '$d$' km. Since the distances are equal, Tanmay covers a total distance of $3d$ km.

We need to calculate the time taken for each segment using the formula:

Time = Distance / Speed

• Time taken for the first distance ($t_1$): $t_1 = \frac{d}{2}$ hours.
• Time taken for the second distance ($t_2$): $t_2 = \frac{d}{3}$ hours.
• Time taken for the third distance ($t_3$): $t_3 = \frac{d}{4}$ hours.

Calculating Total Time and Average Speed

To find the average speed, we first need the total time taken for the entire journey.

Total Time = $t_1 + t_2 + t_3$

Total Time = $\frac{d}{2} + \frac{d}{3} + \frac{d}{4}$

To add these fractions, we find a common denominator, which is the Least Common Multiple (LCM) of 2, 3, and 4. The LCM(2, 3, 4) = 12.

Total Time = $\frac{6d}{12} + \frac{4d}{12} + \frac{3d}{12}$

Total Time = $\frac{6d + 4d + 3d}{12}$

Total Time = $\frac{13d}{12}$ hours.

Now, we can calculate the Average Speed:

Average Speed = Total Distance / Total Time

Average Speed = $\frac{3d}{\frac{13d}{12}}$

Average Speed = $3d \times \frac{12}{13d}$

The '$d$' cancels out:

Average Speed = $\frac{3 \times 12}{13}$

Average Speed = $\frac{36}{13}$ km/h.

Rounding the Average Speed

The question asks for the answer rounded off to 2 decimal places.

$\frac{36}{13} \approx 2.76923...$

Rounding this to two decimal places gives us 2.77.

Therefore, Tanmay's average speed during the whole journey is approximately 2.77 km/h.