পেট্রোবাংলা প্রিলি ও লিখিত সমন্বিত প্রস্তুতি [Archived]

Question: A person crosses a 600 m long street in 5 minutes, What is his speed in km per hour?

Solution:
Speed = 600/(5 × 60) m/sec
= 2 m/sec
= 2 × (18/5) km/hr
= 7.2 km/hr

Question: The speed of a car increases by 2 kms after every one hour. If the distance travelling in the first one hour was 35 kms. what was the total distance travelled in 12 hours?

Solution:
Total distance travelled in 12 hours = (35 + 37 + 39 +..... upto 12 terms)
This is an A.P with first term, a = 35,
number of terms, n = 12
d = 2.

Required distance = (12/2){2 × 35 + (12 - 1) × 2}
= 6 (70 + 22)
= 552 kms.

Question: Two trains A and B start simultaneously in the opposite direction from two points P and Q and arrive at their destinations 16 and 9 hours respectively after their meeting each other. At what speed does the second train B travel if the first train travels at 120 km/h?

Solution:
After meeting with each other the 1st train travels 120 × 16 km=1920km.

If the speed of the 2nd train be X km/h, it travels 1920/X hour before meeting.

After meeting, the 2nd train travels 9X km.

Before meeting, the first train covered this distance in 9X/120 hour.

As the two trains started simultaneously, before meeting their journey time was same.
So,
1920/X = 9X/120
⇒ 9X² = 1920 × 120
⇒ X² = (1920 × 120)/9
⇒ X = √[640 × 40]
∴ X=160.

∴ The train B travels at 160km/h.

Question: A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

Solution:
Relative speed of the thief and policeman = (11 - 10) km/hr = 1 km/hr

Distance covered in 6 minutes = (1/60) × 6 km = 1/10 km = 100 m

Therefore, Distance between the thief and policeman = (200 - 100) m = 100 m.

Question: A man takes 20 minutes to row 12 km upstream which is a third more than the time he takes on his way downstream. What is his speed in still water?

Solution:
Let the speed in still water = x km/hr.
Takes 20 min. to row 12 km upstream 
∴ speed of u/s = 36 km/hr.
Also, time taken for u/s is 1/3 more than for d/s.
∴ distance covered in d/s will be 1/3 more.

Hence distance covered by man for d/s in 20 min. = 12 + (12/3) = 16 km.
So speed of d/s = 48 km/hr.

∴ x + y = 48
and x - y = 36

∴ x + y + x - y = 48 + 36
⇒ 2x = 84
⇒ x = 42 km/hr.

Question: A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed for the first 320 km of the tour is-

Solution:
Total time taken = (160/64 + 160/80)hrs
= (5/2 + 2)
= 9/2 hrs.

∴ Average speed = (320 × (2/9) km.hr
= 71.11 km/hr.

Question: A train moving at speed of 90 km/hr crosses a pole in 7 seconds. Find the length of the train.

Solution:
Length of the train is equal to the distance covered by train to cross the pole. So, we will find the distance travelled by the train in 7 seconds by applying the following formula:
Distance = Speed × Time

Speed is given in Km/hr so we will convert it into m/s as answers are given in meters.
Speed = 90 × (5/18) = 25 m/s

Time = 7 seconds
∴ Distance = 25 × 7= 175 meters

Question: How long will a boy take to run round a square field of side 35 meters, If he runs at the rate of 9 km/hr?

Solution:
Speed = 9 km/hr = 9 × (5/18) m/sec = 5/2 m/sec

Distance = (35 × 4) m = 140 m.

Time taken = 140 × (2/5) sec= 56 sec

Question: How long will it take to row 20 km upstream if one can row 10 km in 10 minutes in still water and the same distance in 8 minutes with the stream?

Solution:
Let x be the speed of man in still water
and y be the speed of stream.
∴ Speed of man (x) = 60 km/hr
and speed of downstream = 75 km/hr.
∴ Speed of stream = 15 km/hr.

Hence upstream speed = 60 - 15 = 45 km/hr.

So time taken to cover 20 km = (20/45) × 60 min = 26.67min.

Question: A man reaches his office 20 min late, if he walks from his home at 3 km per hour and reaches 30 min early if he walks 4 km per hour. How far is his office from his house?

Solution:
Let distance = x km.
Time taken at 3 kmph : dist/speed = x/3 = 20 min late.
time taken at 4 kmph : x/4 = 30 min earlier
difference between time taken : 30 - (- 20) = 50 mins = 50/60 hours.
Here,
x/3 - x/4 = 50/60
⇒ x/12 = 5/6
∴ x = 10 km.

Question: A train of length 200 meters crosses a man running at 10 km/hr in the same direction in 10 seconds. What is the speed of the train?

Solution:
When the train and man are moving in same direction then relative speed will be the difference between their individual speeds. In this problem the other way to find the relative speed is to divide the distance covered (length of train) by the time taken by the train to cross the man.

Relative Speed = 200/10

We will convert it into Km/hr
(200/10) × (18/5) = 72 km/hr

Now, let the speed of the train is X km/hr.
So, the relative speed, 72 km/hr = X km/hr - 10 km/hr
⇒ X - 10 = 72
⇒ X = 72 + 10
∴ X = 82 km/hr

Question: Hemal and Titu start walking towards each other at 4 AM at the speed of 5 kmph and 3 kmph respectively. They were initially 28 km apart. At what time do they meet?

Solution:
Let both will meet after t hours
Distance covered by Hemal in t hours at a speed of 5kmph = 5t
Distance covered by Titu in t hours at a speed of 3kmph = 3t
Total Distance = 28 km

According to question, 
5t + 3t = 28
⇒ 8t = 28
⇒ t = 28/8 = 3.5 =  3hours 30mins

∴ Hemal and Titu started at 4 am so they will meet after 3 hours 30mins at 7 : 30 AM

Question: A boat makes a return journey from point A to point B and back in 5 hours 36 minutes. One way it travels with the stream and on the return it travels against the stream. If the speed of the stream increases by 2 km/hr, the return journey takes 9 hours 20 minutes. What is the speed of the boat in still water? (The distance between A and B is 16 km.)

Solution:
Let x be speed of u/s
and y be the speed of d/s.

∴ (16/x) + (16/y) = (28/5)
and 16/(y+2) + 16/(x-2) = 28/3

Solving these 2 equations,
we get x = 4km/hr
and y = 10km/hr

∴ speed of boat in still water = (4 + 10)/2 = 7 km/hr.

Question: In a 1500 m race, Akash beats Bakul by 150 m and in the same race Bakul beats Chadni by 75 m. By what distance does Akash beat Chadni?

Solution:
According to the question,
Akash goes 1500 m while Bakul goes (1500 - 150) i.e. 1350m.
Ratio of speed of Akash and Bakul = 1500 : 1350 = 10 : 9 = 200 : 180

According to the question,
Bakul goes 1500 m while Chadni goes (1500 - 75) i.e. 1425m.
Ratio of speed of Bakul and Chadni = 1500 : 1425 = 20 : 19 = 180 : 171

So, the ratio of the speeds of Akash, Bakul and Chadni = 200 : 180 : 171
Let the speeds of Akash, Bakul and Chadni be 200k, 180k and 171k m/s respectively.

Time taken by Akash to finish the race = 1500/200k = 7.5/k seconds
Now,
Akash beats Chadni by = (200 - 171)k × (7.5/k) = 217.5 m

∴ Akash beat Chadni by 217.5m.

Question: A train moving at 50 km/hr crosses a bridge in 45 seconds. The length of train is 150 meters. Find the length of the bridge.

Solution:
The distance covered by train when it crosses the bridge is equal to the sum of length of the train and length of the bridge.

Speed of train in m/s= 50 × (5/18) m/s = 125/9 m/s
Time = 45 seconds

Let the length of the bridge is X.

Speed of the train = (length of train + length of bridge)/time taken to cross the bridge
⇒ 125/9 = (150 + X)/45
⇒ 125 × 45 = 9(150 + X)
⇒ 625 = 150 + X
∴ X = 475

Question: An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 5/3 hours, it must travel at a speed of-

Solution:
Distance = (240 × 5) = 1200 km.
Speed = Distance/Time
Speed = 1200/(5/3) km/hr.

∴ Required speed = 1200 × (3/5) km/hr = 720 km/hr.

Question: A man rows 24 km upstream in 6 hours and a distance of 35 km downstream in 7 hours. Then the speed of the man in still water is-

Solution:
Speed of upstream = 24/6 = 4 km/hr.
Speed of downstream = 35/7 = 5 km/hr.

∴ Speed of man in still water = (4 + 5)/2 = 4.5 km / hr.

Question: If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. The actual distance travelled by him is-

Solution:
Let the actual distance travelled be x km.
Then,
x/10 = (x + 20)/14
⇒ 14x = 10x + 200
⇒ 4x = 200
∴ x = 50 km.

Question: A train is moving at 120 km/hr. The length of the train is 150 meters. How long it will take to cross a platform of length 100 meters?

Solution:
Speed of train in m/s = 120 × (5/18) m/s = 100/3 m/s
Distance covered to cross the platform is equal to the sum of length of the train and length of the platform.
So, distance= 150 + 100 = 250 meters

Time = distance/speed
= 250 × (3/100)
= 7.5 sec

Question: Excluding stoppages, the speed of a bus is 54 kmph and including stoppages, it is 45 kmph. For how many minutes does the bus stop per hour?

Solution:
Due to stoppages, it covers 9 km less.

Time taken to cover 9 km = (9/54) × 60 min = 10 min.

Question: A boat goes 12 km upstream in 48 minutes. The speed of stream is 2 km/hr. The speed of boat in still water is-

Solution:
12 km upstream in 48 min.
⇒ it will cover 15 km in 1 hr.

Speed of stream = 2 km / hr.

∴ Speed of boat in still water = 15 + 2 = 17 km/hr.

Question: In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. The duration of the flight is-

Solution:
Let the duration of the flight be x hours.
Then,
600/x - 600/(x + 1/2) = 200
⇒ 600/x - 1200/(2x + 1) = 200
⇒ 3/x - 6/(2x + 1) = 1
⇒ (6x + 3 - 6x){x(2x + 1)} = 1
⇒ x(2x + 1) = 3
⇒ 2x² + x - 3 = 0
⇒ 2x² + 3x - 2x - 3 = 0
⇒ x(2x + 3) - 1(2x + 3) = 0
⇒ (2x + 3)(x - 1) = 0
∴ x = 1 hr. [neglecting the (- ve) value of x]

Question: A train of length 100 meters is moving at a speed of 70 km/hr. In what time it will cross a man who is walking at 10 km/hr in the same direction?

Solution:
n this problem, both the train and man are moving so we will find the relative speed of the train. They are moving in the same direction, so the relative speed = (speed of train - speed of man)

Relative Speed = (70-10) = 60 km/hr

Relative Speed in m/s = 60 × (5/18) = 50/3 m/s

Distance covered to cross the man = length of the train (100 meters)

Time = 100 × (3/50) sec
= 6 sec

Question: A man complete a journey in 10 hours. He travels first half of the journey at the rate of 21 km/hr and second half at the rate of 24 km/hr. Find the total journey in km.

Solution:
Let,
Total distance = x

{(1/2)x}/21 + {(1/2)x}/24 = 10
⇒ x/21 + x/24 = 20
⇒ 15x = 168 × 20
⇒ x = (168 × 20)/15
∴ x  = 224 km.

Question: A motorboat can travel at 5 km/hr in still water. It travelled 90 km downstream in a river and then returned, taking altogether 100 hours. Find the rate of flow of the river.

Solution:
Speed of boat in still water = x = 5 km/hr.
Let rate of flow of river = y km/hr.
∴ Speed of u/s = 5 - y
and speed of d/s = 5 + y

∴ 90/(5 + y) + 90/(5 - y) = 100
⇒ 450 - 9y + 450 + 9y = 100(25 - y²)
⇒ 9 = 25 - y²
⇒ y² = 16
⇒ y = 4 km/hr.

Question: In covering a distance of 30 km, Abhay takes 2 hours more than Sameer. If Abhay doubles his speed, then he would take 1 hour less than Sameer. Abhay's speed is-

Solution:
Let Abhay's speed be x km/hr.
Then,
30/x - 30/2x = 3
⇒ 6x = 30
∴ x = 5 km/hr.

Question: The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 km in 4 hours, then the speed of the first train is-

Solution:
Let the speed of two trains be 7x and 8x km/hr.
Then,
8x = (400/4) = 100
∴ x = 100/8 = 12.5

Speed of first train = (7 × 12.5) km/hr = 87.5 km/hr.

Question: Robert is travelling on his cycle and has calculated to reach point A at 2 P.M. if he travels at 10 kmph, he will reach there at 12 noon if he travels at 15 kmph. At what speed must he travel to reach A at 1 P.M.?

Solution:
Let the distance travelled by x km.
Then,
x/10 - x/15 = 2
⇒ 3x - 2x = 60
∴ x = 60 km.

Time taken to travel 60 km at 10 km/hr = 60/10 hrs = 6 hrs.
So, Robert started 6 hours before 2 P.M. i.e., at 8 A.M.

∴ Required speed = 60/5 kmph. = 12 kmph.

Question: A boatman can row 2 km against the stream in 20 minutes and return in 10 minutes. Find the rate of flow of the current.

Solution:
Let,
x be the speed of man in still water
and y be the speed of current.

Speed of d/s = (2/10) × 60 = 12 km/hr.
Speed of u/s = (2/20) × 60 = 6 km/hr.

∴ rate of current = (12 - 6)/2 = 3 km/hr.

Question: Two trains of length 140 meters and 166 meters are moving towards each other on parallel tracks at a speed of 50 km/hr and 60 km/hr respectively. In what time the trains will cross each other from the moment they meet?

Solution:
In this problem, both the trains are moving so we will find the relative speed of the train. They are moving in the opposite direction, so the relative speed will be sum of their individual speeds.

Relative speed: (50 + 60) = 110 km/hr

Relative speed in m/s = 110 × (5/18) m/s = 550/18 m/s = 275/9 m/s

Distance covered is equal to the sum of the length of trains: 140 + 166= 306 meters

Time = 306 × (9/275) sec
= 10.01 sec
≈ 10 sec