ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: A person walks at 14 km/h instead of 10 km/h and in the same time covers 20 km more distance. Find the actual distance traveled.

Solution: 
Given that,
Usual speed = 10 km/h
Increased speed = 14 km/h
Extra distance covered = 20 km

Let the actual distance traveled be = d km

Now,
Time = d/10 
And, At 14 km/h, distance covered in the same time = d + 20 
So,
⇒ d/10 = (d + 20)/14
⇒ 14d = 10d + 200
⇒ 14d - 10d = 200
⇒ 4d = 200
⇒ d = 200/4
∴ d = 50 km

The actual distance traveled is 50 km.

Question: A train is 100 meter long and is running at the speed of 30 km per hour. Find the time it will take to pass a man standing at a crossing.

Solution: 
Given that, 
Length of train, L = 100 meters
Speed of train, v = 30 km/h
= 30 × (1000/3600) m/s
= 30 × (5/18) m/s
= 150/18 m/s 
= 25/3 m/s

We know, 
Time taken = Distance/Speed
= 100/(25/3) m/s
= 100 × (3/25)
= 300/25
= 12 seconds

So the train takes 12 seconds to pass the man standing at the crossing.

Question: A boat can cover 'r' km upstream in 6 hours. It can cover 'r + 18' km downstream in 4 hours. Find the time taken by boat to cover 81 km upstream and 90 km downstream if the speed of boat in still water and speed of stream are in the ratio 4 : 1, respectively.

Solution:
Given that,
Boat covers r km upstream in 6 hours
Boat covers (r + 18) km downstream in 4 hours
Ratio of speed of boat in still water : speed of stream = 4 : 1

Let speed of boat in still water = 4k km/hr
Let speed of stream = 1k km/hr

∴ Upstream speed = 4k − 1k = 3k
∴  Downstream speed = 4k + 1k = 5k

Now,
r/6 = 3k
∴ r = 18k …… (1)

And, (r + 18)/4 = 5k
(18k + 18)/4 = 5k ; [From 1]
⇒ 18k + 18 = 20k
⇒ 20k - 18k = 18
⇒ 2k = 18
⇒ k = 18/2 = 9
∴ k = 9

Now, upstream speed = 3k = 27 km/hr
Downstream speed = 5k = 45 km/hr

Required Time,
Time for 81 km upstream = 81/27 = 3 hours
Time for 90 km downstream = 90/45 = 2 hours

∴ Total time = 3 + 2 = 5 hours

∴ The boat takes 5 hours to cover 81 km upstream and 90 km downstream.

Question: A man travels from his home to office at 4km/hr and reaches his office 20 min late. If the Speed had been 6 km/hr he would have reached 10 min early. Find the distance from his home to office?

Solution:
Let the distance from home to office = d km

Time taken at 4 km/h = d/4 hours
And time taken at 6 km/h = d/6 hours

The difference between these two times = late time + early time = 20 min + 10 min = 30 min = 1/2 hours

ATQ, 
(d/4) - (d/6) = 1/2
⇒ (3d - 2d)/12 = 1/2
⇒ d = 12/2
∴ d = 6 km

So the distance from home to office is 6 km.

Question: A train running at the speed of 60 km/hr crosses a pole in 9 seconds. Find the length of the train.

Solution: 
Given that, 
Speed of train, v = 60 km/h
Time to cross a pole, t = 9 seconds

We know, 
Length of the train = speed × time
= (50/3) × 9
= (50 × 9)/3
= 450/3
= 150 meters

So the length of the train is 150 meters.

Question: A boat takes 6 hours to travel 30 km upstream and 3 hours to travel the same distance downstream. Find the distance travelled by the boat in 7 hours in still water.

Solution: 
Let the speed of the boat in still water be b km/h
and the speed of the current (stream) be c km/h.
Then we get,
Upstream speed = b - c
Downstream speed = b + c

Now, upstream speed = 30/6 = 5 km/h
∴ b - c = 5  ……… (1)
And downstream speed = 30/3 = 10 km/h
∴ b + c = 10 ……… (2)

Add equations (1) and (2) then we get,
(b - c) + (b + c) = 5 + 10
⇒ 2b = 15
⇒ b = 15/2
∴ b = 7.5 km/h
So, the speed of the boat in still water is 7.5 km/h.

∴ Distance travel in still water in 7 hours = speed × time
= 7.5 × 7
= 52.5 km

So the boat will travel 52.5 km in 7 hours in still water. 

Question: A car travels at a speed that is 3/4th the speed of a bike. The bike covers 240 km in 4 hours. How much distance will the car cover in 30 minutes?

Solution:
Speed of the bike = distance/time
= 240/4 = 60 km/h

And speed of the car = 3/4 of the speed of the bike
= (3/4) × 60 = 45 km/h

∴ Time for the car = 30 minutes
= 30/60 = 1/2 hours

∴ Distance covered by the car = speed × time
= 45 × (1/2) = 22.5 km

So the car will cover 22.5 km in 30 minutes.

Question: From P and Q, two trains start moving towards each other at the same time. Their speeds are 120 km/h and 100 km/h, respectively. When the two trains meet each other, one train has covered 40 km more than other train. Find the distance between P and Q?

Solution: 
Speeds are in the ratio 120 : 100 = 6 : 5
So distances covered in the same time are also in the ratio 6 : 5

Let distances be 6k and 5k.

∴ Difference = 6k - 5k = 40 
∴ k = 40

∴ Total distance = 6k + 5k = 11k = 11 × 40 = 440 km

So the distance between P and Q is 440 km.

Question: A boat goes 20 km upstream and 44 km downstream in 8 hours. In 5 hours, it goes 15 km upstream and 22 km downstream. Determine the speed of the boat in still water.

Solution: 
Let,
Upstream speed = U km/h
Downstream speed = D km/h

Then we get speed of boat = (U + D)/2

Now,
According to the question,
20/U + 44/D = 8 ....… (i)
15/U + 22/D = 5 ....… (ii)

Now, multiply by 2 the equation (ii) then subtract from equation (i) we get
20/U + 44/D = 8
30/U + 44/D = 10
⇒ - 10/U = - 2
⇒ 2U = 10
⇒ U = 10/2 = 5
∴ U = 5 km/hr

Putting the value in equation (i), we get
20/5 + 44/D = 8
⇒ 44/D = 8 - 4
⇒ 4D = 44
⇒ D = 44/4
∴ D = 11

So, the speed of boat = (U + D)/2 = (5 + 11)/2 = 8 km/hr

So the speed of the boat in still water is 8 km/h.

Question: A bus travels the first 150 km at 75 km/h and the next 150 km at 50 km/h. What is the average speed for the whole trip?

Solution: 
Total distance = 150 km + 150 km = 300 km

Now, 
Time for first part, 
Time = 150/75 = 2 hours

And time for second part, 
time = 150/50 = 3 hours

∴ Total time = 2 + 3 = 5 hours

Average speed = Total distance/Total time
= 300/5
= 60 km/h

So the average speed for the whole trip is 60 km/h.

Shortcut:
When equal distances are covered at speeds a and b,
Average speed = 2ab/(a + b)
= (2 × 75 × 50)/(75 + 50) 
= 60 km/h

Question: A train travels between X and Y in 3 hours. When the speed of train is increased by 6 km/hr, then it covers the same distance in 2 hours. What is the original speed of train?

Solution: 
Let the original speed of the train be a km/hr.
The distance between X and Y is the same in both cases.

Now, given that, 
Speed = a km/hr
Time = 3 hours
∴ Distance = a × 3 = 3a km

And,
Speed = (a + 6) km/hr
Time = 2 hours
∴ Distance = (a + 6) × 2 = 2(a + 6) km

Since distance is the same. Then we get,
⇒ 3a = 2(a + 6)
⇒ 3a = 2a + 12
⇒ 3a - 2a = 12
∴ a = 12

So the original speed of the train is 12 km/hr

Question: A man rows a boat a certain distance downstream in 9 hours, while it takes 18 hours to row the same distance upstream. How many hours will it take him to row three-fifth of the same distance in still water?

Solution: 
Let distance = d
So downstream speed = d/9​
And upstream speed = d/18​

We know, 
Speed in still water = (Downstream + Upstream)/2
= (d/9 + d/18)/2 
= {(2d + d)/18}/2
= (d/6)/2
= d/12
So, still water speed = d/12
Distance to travel = 3d/5

∴ Time = (3d/5)/(d/12) = (3 × 12)/5 = 36/5 = 7.2 hours

So it will take the man 7.2 hours to row three-fifth of the distance in still water. 

Question: A truck travels at 96 km/h. How much distance will it cover in 75 minutes?

Solution:
Given that,
Speed = 96 km/h
Time = 75 minutes = 75/60 hours = 5/4 hours

We know,
Distance = Speed × Time
= 96 × (5/4)
= 96 × 1.25
= 120 km

So the truck will cover 120 km in 75 minutes.

Question: A train passes a stationary pole in 8 seconds. The train also passes a 200 m long bridge in 28 seconds. What is the length of the train?

Solution: 
Given that,
Time to pass a pole = 8 s
Time to pass a 200 m bridge = 28 s

Let the length of the train = L meters
When passing a pole, the train covers distance = L in 8 s
And when passing a bridge, distance = (L + 200) in 28 s

Now, 
From pole,
Speed = Distance/Time = L/8 m/s 

And, 
From bridge,
Speed = Distance/Time = (L + 200)/28 m/s

ATQ, 
L/8 = (L + 200)/28
⇒ 28L = 8L + 1600
⇒ 28L - 8L = 1600
⇒ 20L = 1600
⇒ L = 1600/20
∴ L = 80 m

So the length of the train is 80 meters. 

Question: A boat's speed with the current is 17 km/hr and the speed of the current is 3.5 km/hr. What is the boat's speed against the current?

Solution:
Given that, 
Boat's speed with the current = 17 km/hr
Speed of the current = 3.5 km/hr

∴ Boat's speed in still water = speed with the current - speed of the current
= (17 - 3.5) km/hr
= 13.5 km/hr

∴ Boat's speed against the current = speed in still water - speed of the current
= (13.5 - 3.5) km/hr
= 10 km/hr

Therefore, the boat's speed against the current is 10 km/hr.

Question: A car reaches from City A to City B in 9 hours travelling at a speed of 40 km/hr. If its speed is increased by 20 km/hr, then the time of journey is reduced by-

Solution: 
Original speed = 40 km/hr
Time taken = 9 hours

∴ Distance between City A and City B = speed × time
= 40 × 9 = 360 km

∴ New speed = 40 + 20 = 60 km/hr

∴ New time taken = distance/new speed
= 360/60
= 6 hours

∴ Reduction in time = original time - new time
= 9 - 6
= 3 hours

So the time of journey is reduced by 3 hours.

Question: Two trains with lengths 126 m and 119 m respectively are moving towards each other. Their speeds are 12 m/s and 23 m/s, respectively. What will be the time needed by the trains to cross each other?

Solution:
To cross each other completely, the two trains must cover a total distance equal to the sum of their lengths.
∴ Total distance to be covered = 126 + 119 = 245 meters

Since they are moving towards each other, their relative speed is the sum of their individual speeds.
∴ Relative speed = 12 + 23 = 35 m/s

∴ Time taken to cross each other = Total distance/Relative speed
= 245/35
= 7 seconds

∴ The trains will take 7 seconds to completely cross each other.

Question: A swimmer swims from a point P against the current for 6 min and then swims back along the current for next 6 min and reaches at a point Q. If the distance between P and Q is 120 m then the speed of the current (in km/h) is:

Solution: 
Given that, 
Swimmer swims 6 min against current and 6 min along current
Distance between P and Q = 120 m

Let,
Speed of swimmer in still water = u m/min
Speed of current = v m/min

Now,
Distance travelled upstream (against current), d₁ = (u - v) × 6
And distance travelled downstream (along current), d₂ = (u + v) × 6

Net displacement from starting point = 120 m
d₂ - d₁ = 120
⇒ 6(u + v) - 6(u - v) = 120
⇒ 6u + 6v - 6u + 6v = 120 
⇒ 12v = 120
⇒ v = 120/12
⇒ v = 10 m/min
⇒ v = (10 × 60)/1000
⇒ v = 600/1000
∴ v = 0.6 km/h

So the speed of the current is 0.6 km/h.

Question: P, Q and R are in a cycle race of 4500 meters. P cycles twice as fast as Q. R cycles 1/3 as fast as Q. R completes the race in 45 minutes. Then where was Q from the finishing line when P finished the race?

Solution:
Given that, 
Race = 4500 m
R finishes in 45 min
∴ speed of R = 4500/45 = 100 m/min

R’s speed = (1/3) Q’s speed
 ∴ Q’s speed = 300 m/min

And, P’s speed = 2 × Q’s speed = 600 m/min

∴ Time for P to finish 4500 m = 4500/600 = 7.5 min
In 7.5 min, Q covers = 300 × 7.5 = 2250 m

∴ Distance left for Q = 4500 - 2250 = 2250 m

So Q was 2250 meters from the finishing line when P finished.

Question: Two trains A and B are moving in the same direction. A has speed of 8 km/h and B has speed of 13 km/h. What is relative speed of B with respect to A?

Solution: 
Given that,
Speed of train A = 8 km/h
Speed of train B = 13 km/h

Since both are moving in the same direction, and B is faster.
∴ Relative speed of B with respect to A = Speed of B - Speed of A
= 13 km/h - 8 km/h
= 5 km/h

So the relative speed of B with respect to A is 5 km/h.

Question: The speed of a boat down the stream is 125% of the speed in still water. If the boat takes 30 minutes to cover 20 km in still water, then how much time (in hours) will it take to cover 15 km upstream?

Solution: 
Given that, 
Speed downstream = 125% of speed in still water
Distance in still water = 20 km, time = 30 min = 0.5 hr
Distance to travel upstream = 15 km

Speed of boat in still water = Distance/Time = 20/0.5 = 40 km/h
And,
Downstream speed = 125% of still water speed.
∴ Downstream speed = 1.25 × 40 = 50 km/h 

We know,
Downstream speed = Boat speed in still water + Current speed 
⇒ 50 = 40 + Current speed 
⇒  Current speed = 50 - 40 = 10 km/h 

And upstream speed = Boat speed in still water - Current speed = 40 - 10 = 30 km/h 

∴ Time to cover 15 km upstream = Distance/Speed ​= 15/30 = 1/2 = 0.5 hours

So the time required for the boat to cover 15 km upstream is 0.5 hours. 

Question: The ratio between the speeds of two buses is 5 : 6. If the second bus runs 450 km in 5 hours, then the speed of the first bus is:

Solution:
Given that,
Ratio of speeds of first bus : second bus = 5 : 6
Second bus covers 450 km in 5 hours

Now, Speed of second bus = distance/time
= 450/5
= 90 km/h

Let speed of first bus = 5x km/h
Speed of second bus = 6x km/h
We know speed of second bus = 90 km/h
So, 6x = 90
⇒ x = 90/6
∴ x = 15

∴ Speed of first bus = 5x = 5 × 15 = 75 km/h

So the speed of the first bus is 75 km/h.

Question: A train is moving at a speed of 132 km/hour. If the length of the train is 110 meters, how long will it take to cross a railway platform 165 meters long.

Solution: 
Given that,
Length of train = 110 m
Length of platform = 165 m
∴ Total distance to be covered = 110 + 165 = 275 meters
Speed of train = 132 km/h
= 132 × (1000/3600) m/s
= 132 × (5/18) m/s
= 110/3 m/s

Time taken = Distance/Speed
= 275/(110/3)
= (275 × 3)/110 
= 7.5 seconds

So the train will take 7.5 seconds to cross the 165 meter long railway platform.

Question: A boat goes 10 km upstream in 50 minutes, and the speed of the stream is 3 kmph. Find the speed of the boat in still water (in km/h). 

Solution: 
Let the speed of the boat in still water = x km/h
Speed of the stream = 3 km/h (given)
Upstream speed = x - 3 km/h
Distance upstream = 10 km
Time upstream = 50 minutes = 50/60 hours = 5/6 hours

We know,
Speed = Distance/Time
Upstream speed = 10 /(5/6) = 10 × (6/5) = 12 km/h
So,
⇒ x - 3 = 12
⇒ x = 12 + 3
∴ x = 15 km/h

The speed of the boat in still water is 15 km/h.

Question: Excluding stoppages, a train travels at 72 km/h and including stoppages, its average speed is 60 km/h. For how many minutes does the train stop per hour?

Solution:
At its moving speed of 72 km/h, the time required to cover 60 km is:
∴ Time moving = distance/speed
= 60 km/72 km/h
= 60/72 hours
= 5/6 hours
= (5/6) × 60 minutes
= 50 minutes

We know,
 1 hour = 60 minutes

∴ Stopping time = 60 - 50 = 10 minutes

∴ The train stops for 10 minutes per hour.