To find the average speed for the whole journey when the speeds for the two parts of the journey are different, we use the harmonic mean of the two speeds, not the arithmetic mean. The formula for the average speed in such cases is:

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

Given:

• Speed from $A$ to $B$ ($v_1$) = $2 \, \text{km/h}$
• Speed from $B$ to $A$ ($v_2$) = $4 \, \text{km/h}$

Solution:

Substitute the values into the formula:

$\text{Average Speed} = \frac{2 \cdot 2 \cdot 4}{2 + 4} = \frac{16}{6} = 2.\overline{6} \, \text{km/h}$

Thus, the average speed for the whole journey is approximately:

$\boxed{2.67 \, \text{km/h}}$

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Related Questions:

1. How is harmonic mean different from arithmetic mean in speed problems?
2. What would the average speed be if the speeds were $3 \, \text{km/h}$ and $6 \, \text{km/h}$?
3. How is the average speed affected by unequal travel distances versus equal distances?
4. Why do we use harmonic mean for average speed in round trips?
5. What happens to the average speed if both speeds are equal?

Tip:

When dealing with average speed over equal distances at different speeds, always use the harmonic mean formula for accuracy!