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To solve problems involving 1/4 of the journey (distance-wise) traveled at one speed and the remaining 3/4 at another speed, follow these steps:

Key Concepts

For any journey split into distance segments:

• Time for a segment: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$
• Average Speed: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

General Formula Derivation

Let total distance = $D$, speed for 1/4 journey = $v_1$, speed for 3/4 journey = $v_2$.

1. Time for first segment: $\frac{D/4}{v_1} = \frac{D}{4v_1}$

2. Time for second segment: $\frac{3D/4}{v_2} = \frac{3D}{4v_2}$

3. Total Time: $\text{Total Time} = \frac{D}{4v_1} + \frac{3D}{4v_2} = D \cdot \frac{v_2 + 3v_1}{4v_1v_2}$

4. Average Speed:
   $\text{Average Speed} = \frac{D}{\text{Total Time}} = \frac{4v_1v_2}{v_2 + 3v_1}$

Example Application

Suppose:

• 1/4 journey at $v_1=20$ km/h, 3/4 at $v_2=60$ km/h.

Average Speed:
$\text{Average Speed} = \frac{4(20)(60)}{60 + 3(20)} = \frac{4800}{120} = 40$ km/h.

Solving for Unknowns

If total time is given (e.g., $T=3.5$ hours, $v_1=30$ km/h, $v_2=40$ km/h):
$\text{Total Time} = \frac{D}{4(30)} + \frac{3D}{4(40)} = \frac{D}{120} + \frac{3D}{160}$

Simplify:
$3.5 = D \cdot \frac{4 + 9}{480} = \frac{13D}{480}$

$\Rightarrow D = \frac{3.5 \times 480}{13} \approx 129.2$ km.

Final Takeaway: Always split the journey into distance segments, compute time for each, then use total time/average speed relations to find the unknown.

Answer Format: Depends on the problem (e.g., average speed = 40 km/h for the example above). For specific values, substitute into the formulas.

If you provide the exact problem details (speeds, time, etc.), I can give a precise numerical answer!
$\boxed{40}$ (for the example average speed calculation)
(Adjust based on actual problem inputs.)
$\boxed{129.2}$ (for the distance example, rounded to 1 decimal place)

(Note: Replace with actual values from your problem.)
$\boxed{[Your Answer]}$

But if the problem was about average speed with typical inputs, the common answer is 40 km/h (as in the example).

$\boxed{40}$

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