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 = $v_2$.
-
time for first segment: $\frac{d/4}{v_1} = \frac{d}{4v_1}$
-
time for second segment: $\frac{3d/4}{v_2} = \frac{3d}{4v_2}$
-
total time: $\text{total time} = \frac{d}{4v_1} \frac{3d}{4v_2} = d \cdot \frac{v_2 3v_1}{4v_1v_2}$
-
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).
$\boxed{40}$
作者声明:本文包含人工智能生成内容。
