Understanding Average Speed

The question asks for the average speed of a scooter that travels from point P to point Q and then back from Q to P at different speeds. It's important to remember that average speed is not simply the average of the two speeds. It is calculated as the total distance traveled divided by the total time taken.

The scooter travels from P to Q and then from Q back to P. This means the distance covered in both directions is the same. Let's assume the distance between point P and point Q is d kilometers.

Calculating Time Taken for Each Leg

The scooter travels from P to Q at a speed of 40 km/h. The time taken for this journey ($t_1$) can be calculated using the formula: Time = Distance / Speed.

$\qquad t_1 = \frac{\text{Distance (P to Q)}}{\text{Speed (P to Q)}} = \frac{d}{40}$ hours

The scooter travels from Q to P at a speed of 30 km/h. The time taken for this return journey ($t_2$) is:

$\qquad t_2 = \frac{\text{Distance (Q to P)}}{\text{Speed (Q to P)}} = \frac{d}{30}$ hours

Calculating Total Distance and Total Time

The total distance traveled by the scooter for the round trip (P to Q and Q to P) is the sum of the distance in each direction:

$\qquad \text{Total Distance} = \text{Distance (P to Q)} + \text{Distance (Q to P)} = d + d = 2d$ kilometers

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

$\qquad \text{Total Time} = t_1 + t_2 = \frac{d}{40} + \frac{d}{30}$ hours

To add the fractions for total time, we find a common denominator, which is 120:

$\qquad \text{Total Time} = \frac{d \times 3}{40 \times 3} + \frac{d \times 4}{30 \times 4} = \frac{3d}{120} + \frac{4d}{120} = \frac{3d + 4d}{120} = \frac{7d}{120}$ hours

Calculating Average Speed

Now we can calculate the average speed using the formula: Average Speed = Total Distance / Total Time.

$\qquad \text{Average Speed} = \frac{2d}{\frac{7d}{120}}$

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

$\qquad \text{Average Speed} = 2d \times \frac{120}{7d}$

The 'd' terms cancel out:

$\qquad \text{Average Speed} = \frac{2 \times 120}{7} = \frac{240}{7}$ km/h

Alternatively, when an object travels the same distance in two parts of a journey with different speeds ($v_1$ and $v_2$), the average speed can be calculated directly using the formula:

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

Substituting the given speeds, $v_1 = 40$ km/h and $v_2 = 30$ km/h:

$\qquad \text{Average Speed} = \frac{2 \times 40 \times 30}{40 + 30} = \frac{2400}{70}$

Simplifying the fraction:

$\qquad \text{Average Speed} = \frac{240}{7}$ km/h

Both methods yield the same result for the average speed of the scooter.

Final Answer

The average speed of the scooter for the round trip is $\frac{240}{7}$ km/h.