Understanding the Average Speed Problem

This question asks us to find the average speed for Dixit's entire journey. The journey has two parts: traveling a certain distance by car and returning the same distance by scooter. We are given the speed for each part of the journey: 28 km/h by car and 36 km/h by scooter.

It's important to remember that when the distance covered is the same for different speeds, the average speed is NOT simply the average of the two speeds. We need to consider the time taken for each part of the journey.

Calculating Average Speed for the Whole Journey

Average speed is defined as the total distance traveled divided by the total time taken.

Let's assume the distance Dixit covered each way is '$d$' km.

• Total Distance: The total distance traveled is the distance going plus the distance returning. Total Distance = $d + d = 2d$ km.
• Time Taken: We need to calculate the time taken for each part separately using the formula: Time = Distance / Speed.
  • Time taken by car (going) = $\frac{d}{28}$ hours.
  • Time taken by scooter (returning) = $\frac{d}{36}$ hours.
• Total Time: Add the time taken for both parts. Total Time = $\frac{d}{28} + \frac{d}{36}$ hours.

To add these fractions, we find a common denominator for 28 and 36. The least common multiple (LCM) of 28 and 36 is 252.

• Total Time = $d \left( \frac{1}{28} + \frac{1}{36} \right)$ Total Time = $d \left( \frac{9}{252} + \frac{7}{252} \right)$ Total Time = $d \left( \frac{9 + 7}{252} \right)$ Total Time = $d \left( \frac{16}{252} \right)$ hours.
• We can simplify the fraction $\frac{16}{252}$ by dividing both numerator and denominator by 4: $\frac{4}{63}$. So, Total Time = $\frac{4d}{63}$ hours.

Final Average Speed Calculation

Now we can calculate the average speed using the formula:

Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$

Average Speed = $\frac{2d}{\frac{4d}{63}}$

To divide by a fraction, we multiply by its reciprocal:

Average Speed = $2d \times \frac{63}{4d}$

The '$d$' in the numerator and denominator cancels out:

Average Speed = $\frac{2 \times 63}{4}$

Average Speed = $\frac{126}{4}$

Average Speed = $31.5$ km/h.

Alternative Method: Harmonic Mean

For a journey covering the same distance at two different speeds ($v_1$ and $v_2$), the average speed can be directly calculated using the harmonic mean formula:

Average Speed = $\frac{2 v_1 v_2}{v_1 + v_2}$

Plugging in the values $v_1 = 28$ km/h and $v_2 = 36$ km/h:

Average Speed = $\frac{2 \times 28 \times 36}{28 + 36}$

Average Speed = $\frac{2016}{64}$

Average Speed = $31.5$ km/h.

Both methods confirm that the average speed for the whole journey is 31.5 km/h.