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Q.

A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. What is the distance covered by the train?

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a

800 km

b

700 km

c

500 km

d

600 km

answer is D.

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Detailed Solution

Given that a train covers a certain distance at a uniform speed.
We know that speed is defined as the distance covered per unit of time.

v = d t or Speed = distance time

.
Here 'v' is speed, 'd' is the distance covered and 't' is the time taken.
Let ‘x’ be the speed and ‘y’ be the time, and distance be 'd',
Therefore,

x = d y

Now let us determine the distance covered from above,
d = xy                      ---- (1)
If the train runs 10 km/h faster, then new speed of train is,
New speed = x + 10        ---- (i)
Also, the train would take 2 hours less than the scheduled time, then the new time is,
New time = y – 2.            ---- (ii)
So the new speed will be,

New speed = distance New time

.
Substituting equation (i) and (ii) in the above equation.

x + 10 = d y − 2

          ---- (2)
If the train was slower by 10km/h, then the new speed will be,
New speed = x – 10.       ---- (iv)
As per the statement, the train would take 3 hours more than the scheduled time, then
New time = y + 3.            ---- (v)
So the new speed will be,

New speed = distance New time

.
Substituting equation (iv) and (v) in the above equation.

x − 10 = d y + 3

---- (3)
Now solving equation (2), we get

⇒ x + 10 = d y − 2 ⇒ (x + 10) (y − 2) = d ⇒ x y − 2 x + 10 y − 20 = d

Substituting equation (1) in the above equation.

⇒ x y − 2 x + 10 y − 20 = x y ⇒ − 2 x + 10 y = 20 ---------(4)

Now solving equation (3),

x − 10 = d y + 3

⇒ (x − 10) (y + 3) = d

⇒ x y + 3 x − 10 y − 30 = d

Putting equation (1) in the above equation.

⇒ x y + 3 x − 10 y − 30 = x y

⇒ 3 x − 10 y = 30

---- (5)
Use the elimination method to determine the value of x and y.
Now add equations (4) & (5).

⇒ − 2 x + 3 x + 10 y + (− 10 y) = 20 + 30 ⇒ x = 50 ----------(6)

Now put equation (6) in equation (4), to get value of ‘y’.

− 2 x + 10 y = 20 ⇒ − 2 (50) + 10 y = 20 ⇒ − 100 + 10 y = 20 ⇒ 10 y = 120 ⇒ y = 12010 ⇒ y = 12 -------(7)

To determine distance, put equations (6) & (7) in equation (1).

d = x y d = (50) (12) d = 600

So, the distance covered by the train is 600km.
Therefore, option 4 is correct.

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