Question

It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car.
It takes 20 minutes more, if 200 km is done by train and the rest by car. What is the
ratio of the speed of the train to that of the car?
3:4
2:3
1:2
1:3​

Answer 1

Solution :-

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

Case (1) :-

120 km by Train , Rest by car, .

→ Distance by car = 600 - 120 = 480km .

And, Time = (Distance / Speed) .

So,

→ (120/x) + (480/y) = 8

→ 120[1/x + 4/y] = 8

→ ( 1/x + 4/y) = (8/120)

→ (1/x + 4/y) = 1/15

Multiplying Both Sides by 15,

→ (15/x + 60/y) = 1 --------------------- Equation (1).

___________________

Case (2) :-

200km By Train, means , 400km by car , and time now is 20min. More .

→ Total time = 8 + 20min . = 8 + (20/60) = 8 + (1/3) = (25/3) Hours.

So,

→ (200/x) + (400/y) = 25/3

→ 25[(8/x) + (16/y)] = 25/3

→ (8/x + 16/y) = 1/3

Multiply by 3 both sides,

→ (24/x + 48/y) = 1 --------------------- Equation (2).

____________________

So,

→ Equation (1) = Equation (2) = 1

→ (15/x + 60/y) = (24/x + 48/y)

→ (60/y - 48/y) = (24/x - 15/x)

→ 12/y = 9/x

→ x/y = 9/12

→ x/y = 3/4

→ x : y = 3 : 4 (Option A) (Ans).

Hence, Ratio of Speed of Train to that of Car is 3:4.

Answer 2

$\large{\underline{\red{\rm{AnswEr=3:4}}}}$

$\huge\bold\purple{\underline{Solution:}}$

$\bold\red{\underline{Let:}}$

The speed of train be X km/h

The speed of car be Y km/h

$\bold\red{\underline{Given\:in\:case\:(i)}}$

Total distance =600km

Total time=8 hours

Now,

Distance covered by train=120 km

Distance covered by car=600-120=480 km

According to 1st case

• Formula used here

$\bold\purple{\boxed{Speed=\frac{Distance}{Time}}}$

⟹120/X+480/Y=8

⟹120Y+480X=8XY------------(1)

$\bold\green{\underline{Given\:in\:case\:(ii)}}$

Total distance=600km

Total time=8hours+20minutes

Total time=25/3 hours

Distance covered by train=200km

Distance covered by car=600-200=400km

Now, according to 2nd case

⟹200/X+400/Y=25/3

⟹200Y+400X=25/3XY-----------(2)

Solving eq 1 and 2 by multiplying and subtracting

⟹24000Y+96000X=1600XY-----(3)

⟹24000Y+48000X=3000/3XY-----(4)

By solving we get here

⟹48000X=600XY

⟹48000=600Y

⟹Y=80

Putting the value of Y=80 in EQ 1

⟹120Y+480X=8XY

⟹120×80+480X=8X×80

⟹9600+480X=640X

⟹640X-480X=9600

⟹160X=9600

⟹X=60

Hence,

The ratio of speed of train and car=60:80=>3:4

$\bold\purple{\boxed{The\: speed\:of\: train=60\:km/h}}$

$\bold\orange{\boxed{The\: speed\:of\: car=80\:km/h}}$