Question

A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.

Answer 1

Let the speed of a train = x km/hr

And the speed of a car = y km/hr

Total distance travelled=600 km

According to the question, if he covers 400 km by train and rest by car i.e. =600-400=200 km

Time taken $=6$ hr 30 min =6.5 hr

If he travels 200 km by train and rest by car i.e (600-200)=400 km

he takes half hour longer i.e 7 hours

So, total time = train time + car time

$\because$ time = distance/speed

$\Rightarrow \frac{400}{x}+\frac{200}{y}=6.5...\left(i\right)$

$\Rightarrow \frac{200}{x}+\frac{400}{y}=\mathrm{7...}\left(ii\right)$

Let take $\frac{1}{x}=u$ and $\frac{1}{y}=v$

$400u+200v=\mathrm{6.5...}\left(iii\right)$

and $200u+400v=\mathrm{7....}\left(iv\right)$

On multiplying equation (iii) by 2 and equation (iv) by 4

we get,

$800u+400v=13$...(a)

$800u+1600v=\mathrm{28...}\left(b\right)$

Now, subtract equation (a) from (b)

$800u+1600v-800u-400v=28-13v=\frac{15}{1200}v=\frac{1}{80}$

On putting the value of $v$ in (iv), we get

$200u+400\times \frac{1}{80}=7$

$200u+5=7u=\frac{1}{100}$

We get,

$x=100,y=80$

Hence, the speed of the train is 100 km/hr and the speed of the car is 80 km/hr.