Calculating Average Speed for a Round Trip

The problem asks us to find the average speed of a person who travels a certain distance at one speed and returns the same distance at a different speed. This is a classic problem involving average speed where the distance covered in each part of the journey is the same.

When a person covers a certain distance at speed \(v_1\) and an equal distance at speed \(v_2\), the average speed for the entire journey is not simply the arithmetic mean \(\frac{v_1 + v_2}{2}\). Instead, we use a different formula because the time taken for each part of the journey is different.

Let the distance covered in one direction be \(d\). The time taken for the outward journey is \(t_1 = \frac{\text{Distance}}{\text{Speed}_1} = \frac{d}{v_1}\). The time taken for the return journey is \(t_2 = \frac{\text{Distance}}{\text{Speed}_2} = \frac{d}{v_2}\).

The total distance covered for the whole journey (out and back) is \(d + d = 2d\). The total time taken is \(t_1 + t_2 = \frac{d}{v_1} + \frac{d}{v_2}\).

The average speed is defined as the total distance divided by the total time.

Average Speed = \(\frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{v_1} + \frac{d}{v_2}}\)

We can simplify the denominator:

\(\frac{d}{v_1} + \frac{d}{v_2} = d \left(\frac{1}{v_1} + \frac{1}{v_2}\right) = d \left(\frac{v_2 + v_1}{v_1 v_2}\right)\)

Now substitute this back into the average speed formula:

Average Speed = \(\frac{2d}{d \left(\frac{v_1 + v_2}{v_1 v_2}\right)} = \frac{2}{\left(\frac{v_1 + v_2}{v_1 v_2}\right)} = \frac{2 v_1 v_2}{v_1 + v_2}\)

This is the formula for the average speed when the distance is the same for both parts of the journey.

In this question, the speed for the outward journey \(v_1 = 60\) kmph, and the speed for the return journey \(v_2 = 40\) kmph.

Let's plug these values into the formula:

Average Speed = \(\frac{2 \times 60 \times 40}{60 + 40}\)

Average Speed = \(\frac{2 \times 60 \times 40}{100}\)

Average Speed = \(\frac{4800}{100}\)

Average Speed = \(48\)

So, the average speed of the person for the whole journey is 48 km/hour.

Let's verify the calculation:

• \(v_1 = 60\) kmph
• \(v_2 = 40\) kmph
• Numerator: \(2 \times 60 \times 40 = 120 \times 40 = 4800\)
• Denominator: \(60 + 40 = 100\)
• Average Speed = \(\frac{4800}{100} = 48\)

The calculated average speed is 48 km/hour.

Comparing this result with the given options:

• 50
• 48
• 42.5
• 45

The calculated average speed matches option 2, which is 48 km/hour.

This confirms that the average speed for the entire journey is 48 km/hour.