Ram's Average Speed Calculation

To determine Ram's average speed for the entire journey, we need the total distance traveled and the total time taken. The average speed is not simply the average of the two speeds because the time spent traveling at each speed is different.

1. Define Distance and Time Variables

Let the distance between Point X and Point Y be $d$ km.

• The total distance for the round trip is $d + d = 2d$ km.
• Time taken to travel from X to Y ($t_1$) = $\frac{\text{Distance}}{\text{Speed}} = \frac{d}{8}$ hours.
• Time taken to return from Y to X ($t_2$) = $\frac{\text{Distance}}{\text{Speed}} = \frac{d}{12}$ hours.

2. Calculate Total Time

The total time for the journey is the sum of the times for each leg:

Total Time = $t_1 + t_2 = \frac{d}{8} + \frac{d}{12}$

Find a common denominator (24) to add the fractions:

Total Time = $\frac{3d}{24} + \frac{2d}{24} = \frac{5d}{24}$ hours.

3. Calculate Average Speed

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

Average Speed = $\frac{2d}{\frac{5d}{24}}$

Simplify the expression:

Average Speed = $2d \times \frac{24}{5d} = \frac{48d}{5d}$

The $d$ cancels out:

Average Speed = $\frac{48}{5}$ km/hr.

4. Final Result

Convert the fraction to a decimal:

Average Speed = $9.6$ km/hr.