Question

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

Answer 1

Case 1 : Total distance $=600km$

distance covered by train $=120km$

time taken by train $=\frac{120}{x}km/h$

$x=$ speed of the train

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

Time taken by car $=\left(\frac{480}{y}\right)km/h$

total time taken $=\frac{120}{x}+\frac{480}{y}=8$

$\frac{30}{x}+\frac{120}{y}=2$

Case II :

total distance covered by train $=200km$

total time taken by train $=\frac{200}{x}$

total distance covered by car $=600-200=400$

total time taken by car $=\frac{400}{y}km/h$

total time taken $=8\times \left(\frac{20}{60}\right)=\frac{25}{3}$

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

$\Rightarrow \frac{600}{x}+\frac{1200}{y}=25$

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

equation are then

$600u+1200v=25$ and

$30u+120v=2$...(1)

Multiplying equation $\left(1\right)$ by $20$ and subtracting the equations we get :

$600u-600u+1200v-2400v=25-40$

$\Rightarrow -1200v=-15$

$\Rightarrow v=\frac{15}{1200}$

Since, $\frac{1}{y}\Rightarrow y=\frac{1200}{15}=80km/h$

speed of car $=80km/h$

multiplying equation $\left(1\right)$ on $10$ and subtracting equations we get

$300u=5$

$u=\frac{5}{300}\Rightarrow \frac{1}{x}=u$

speed of train $=x=\frac{300}{5}=60km/h$