Question : One-third part of a certain journey is covered at the speed of 10 km/hr, one-fourth part at the speed of 15 km/hr, and the rest part at the speed of 20 km/hr. What will be the average speed (in km/hr) for the whole journey?
Option 1: $15$
Option 2: $\frac{200}{17}$
Option 3: $\frac{240}{17}$
Option 4: $\frac{280}{17}$
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Correct Answer: $\frac{240}{17}$
Solution :
Let the total distance of the journey be $d$ km.
One-third of the journey is covered at 10 km/hr, so the time taken is $\frac{d}{3 \times 10} $ hours.
One-fourth of the journey is covered at 15 km/hr, so the time taken is $\frac{d}{4 \times 15}$ hours.
The rest of the journey = $d - \frac{d}{3} - \frac{d}{4} = \frac{5d}{12}$
So, the time taken to cover $ \frac{5d}{12}$ = $\frac{5d}{12 \times 20}$ hours.
The total time taken for the journey is the sum of these times.
The average speed is the total distance divided by the total time.
$\text{Average speed}$
$ = \frac{d}{\left(\frac{d}{3 \times 10} + \frac{d}{4 \times 15} + \frac{5d}{12 \times 20}\right)}$
$= \frac{d}{\left(\frac{d}{30} + \frac{d}{60} + \frac{d}{48}\right)}$
$ = \frac{d}{\frac{17d}{240} }$
$ =\frac{240}{17}$
Hence, the correct answer is $\frac{240}{17}$.
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