Question

Ved travels $600$ km to his home partly by train and partly by car. He takes $8$ hours if he travels $120$km by train and the rest by car. He takes $20$ minutes longer if he travels $200$ km by train and the rest by car. Find the speed of the train and the car.

Answer 1

Let the speed of the train be $x$ km/hr and the speed of the car be $y$ km/hr.

Case I: When he travels $120$ km by train and the rest by car.

If Ved travels $120$km by train, then

Distance covered by car is $(600-120)km=480$km.

Now, Time taken to cover $120$km by train $=\frac{120}{x}$ hrs. $[\because Time=\frac{Distance}{Speed}]$

Time taken to cover $480$ km by car $=\frac{480}{y}$hrs

It is given that the total time of the journey is $8$ hours.

$\therefore \frac{120}{x}+\frac{480}{y}=8$

$\Rightarrow 8(\frac{15}{x}+\frac{60}{y})=8$

$\Rightarrow \frac{15}{x}+\frac{60}{y}=1$

$\Rightarrow \frac{15}{x}+\frac{60}{y}-1=0$ ..(i)

Case II When he travels $200$ km by train and the rest by car

If Ved travels $200$km by train, then

Distance travelled by car is $(600-200)km=400$km

Now, Time taken to cover $200$km by train $=\frac{200}{x}$hrs

Time taken to cover $400$ km by train $=\frac{400}{y}$hrs

In this case the total time of journey is $8$ hour $20$ minutes.

$\therefore \frac{200}{x}+\frac{400}{y}=8$hrs $20$ minutes

$\Rightarrow \frac{200}{x}+\frac{400}{y}=8\frac{1}{3}$ $[\because 8hrs20minutes=8\frac{20}{60}hrs=8\frac{1}{3}hrs]$

$\Rightarrow \frac{200}{x}+\frac{400}{y}=\frac{25}{3}$

$\Rightarrow 25(\frac{8}{x}+\frac{16}{y})=\frac{25}{3}$

$\Rightarrow \frac{8}{x}+\frac{16}{y}=\frac{1}{3}$

$\Rightarrow \frac{24}{x}+\frac{48}{y}=1$

$\Rightarrow \frac{24}{x}+\frac{48}{y}-1=0$ .(ii)

Putting $\frac{1}{x}=u$ and $\frac{1}{y}=v$ in equations (i) and (ii), we get

$15u+60v-1=0$ (iii)

$24u+48v-1=0$ ..(iv)

By using cross-multiplication, we have

$\frac{u}{60\times -1-48\times -1}=\frac{-v}{15\times -1-24\times -1}=\frac{1}{15\times 48-24\times 60}$

$\Rightarrow \frac{u}{-60+48}=\frac{-v}{-15+24}=\frac{1}{720-1440}$

$\Rightarrow \frac{u}{-12}=\frac{v}{-9}=\frac{1}{-720}$

$\Rightarrow u=\frac{-12}{-720}=\frac{1}{60}$ and $v=\frac{-9}{-720}=\frac{1}{80}$

Now, $u=\frac{1}{x}\Rightarrow \frac{1}{60}=\frac{1}{x}\Rightarrow x=60$

and, $v=\frac{1}{y}\Rightarrow \frac{1}{80}=\frac{1}{y}\Rightarrow y=80$

Hence, the speed of a train is $60$ km/hr and the speed of a car is $80$ km/hr.