IBA ফ্যাকাল্টি ভিত্তিক প্রস্তুতি

Question: Eight pipes are fitted to a water tank. Some of these are water pipes to fill the tank and the remaining are waste pipes used to empty the tank. Each water pipe can fill the tank in 12 hours and each waste pipe can empty it in 36 hours. On opening all the pipes an empty tank is filled in 3 hours. The number of waste pipes is-

Solution:
Let,
Number of water pipes = x
So, number of waste pipes = 8 - x
Now,
Total filling rate from water pipes = x × (1/12) = x/12​
Total emptying rate from waste pipes = (8 - x) × (1/36) = (8 - x)/36

ATQ,
⇒ (x/12​) - {(8 - x)/36} = 1/3
⇒ (3x - 8 + x)/36 = 1/3
⇒ 4x - 8 = 12
⇒ 4x = 20
⇒ x = 20/4
∴ x = 5

∴ Number of waste pipes = 8 - x = 8 - 5 = 3

Question: A train takes 15 seconds to pass a stationary point and covers 20 km in 25 minutes. Find the length of the train.

Solution:
Speed = 20km/25min
= (20 × 1000)m/(25 × 60)sec
= (40/3)m/s

∴ Length = Speed × Time to pass point
= (40/3) × 15
= 200 meters

Question: A rower travels at 10 km/hr downstream and 6 km/hr upstream. What are the rower’s speed in still water and the speed of the current?

Solution:
Given that,
Downstream speed = 10 km/h
Upstream speed = 6 km/h

We know that,
Still water speed = (Downstream + Upstream)/2
= (10 + 6)/2 = 8 km/h
And,
Current speed = (Downstream − Upstream)/2
= (10 - 6)/2
= 2 km/h

∴ Speed of still water is 8 km/hr and Speed of the current is 2 km/hr

Question: A policeman sighted a robber from a distance of 300 m. The robber also noticed the policeman and started running at 8 km/hr. The policeman also started running after him at the speed of 10 km/hr. Find the distance that the robber would run before being caught.

Solution:
Since both are running in the same direction, relative speed = 10 - 8 = 2 km/hr 
Now, to catch the robber if he were stagnant, the policeman would have to run 300 m. But since both are moving, the policeman needs to finish off this separation of 300 m. 
= 300 m (or 0.3 km)is to be covered at the relative speed of 2 km/hr. 

∴ Time taken = 0.3/2 = 0.15 hours

Therefore, distance run by robber before being caught = 8 × 0.15 = 1.2 km 

Question: Two pipes, Pipe X and Pipe Y, can fill a tank in 15 minutes and 30 minutes, respectively. If both pipes are opened together, how long will it take to fill the tank?

Solution:
pipe X fill a tank in 15 minutes so, it fills in one minute (1/15)
pipe Y fill a tank in 30 minutes so, it fills in one minute (1/30)

∴ Both pipes fill in one minute = (1/15) + (1/30) = (2 + 1)/30 = 3/30 = 1/10

so, it will take 1/(1/10) or 10 minutes to fill the tank.

Question: A train 120 meters long travels at a speed of 54 km/hr. How long will it take to cross a platform 180 meters long?

Solution:
Total distance = 120 + 180 = 300 meters

∴ Speed = 54 × (5/18) =15 m/s

∴ Time = 300/15 = 20 seconds

Question: A motorboat takes 3 hour to travel 12 km upstream and 12 km downstream in a river with a current of 3 km/hr. What is the boat’s speed in still water?

Solution:
Let, still water speed = x
Then,
Upstream speed = x - 3
Downstream speed = x + 3

ATQ,
⇒ {12/(x - 3)} + {12/ (x + 3)} = 3
⇒ 12(x - 3 + x + 3)/(x² - 9) = 3
⇒ 24x = 3x² - 27
⇒ x² - 8x - 9 = 0
⇒ x² - 9x + x - 9 = 0
⇒ x(x - 9) + 1(x - 9) = 0
⇒ (x - 9)(x + 1) = 0
⇒ x = 9, - 1
∴ x = 9 (positive value only valid)

So the speed of the motorboat in still water is 9 km/hr.

Question: A car travels at a speed of 80 km/hr and reaches its destination in 6 hours. A bus travels at a speed of 50 km/hr and reaches its destination in 10 hours. What is the ratio of the distance covered by the car to the distance covered by the bus?

Solution:
Car,
Speed = 80 km/hr
Time = 6 hours
∴ Distance = 80 × 6 = 480 km

And Bus,
Speed = 50 km/hr
Time = 10 hours
∴ Distance = 50 × 10 = 500 km

∴ Ratio of distances (Car : Bus) = 480 : 500 = 24 : 25

Quesation: A pipe can fill a cistern in 45 minutes while another pipe can empty it in 1 hour 30 minutes. If both pipes are opened at 8 : 15 A.M., at what time will the cistern be full?

Solution:
1 মিনিটে পূর্ণ হয় = 1/45 অংশ
আবার,
1 মিনিটে খালি হয় = 1/(60 + 30) = 1/90 অংশ

∴ 1 মিনিটে পূর্ণ হয় = {(1/45) - (1/90)} = (2 - 1)/90 = 1/90 অংশ

∴ 1/90 অংশ পূর্ণ হয় = 1 মিনিটে
1 বা সম্পূর্ণ অংশ পূর্ণ হয় = 1/(1/90) = 90 মিনিটে

সুতরাং ট্যাংকটি পূর্ণ হবে = 8 : 15 A. M + 90 মিনিটে = 9 : 45 A. M

Question: P can do a piece of work in 10 days and Q can do the same piece of work in 15days. P and Q together complete the same piece of work and get Tk. 1000 as the combined wages. Q's share of the wages will be-

Solution:
P এর 1 দিনের কাজ = 1/10
Q এর 1 দিনের কাজ = 1/15

∴ (P + Q) একত্রে 1 দিনের কাজ = (1/10) + (1/15) = (3 + 2)/30 = 5/30 = 1/6

∴ (P + Q) এর 1 দিনের কাজ অনুপাত = (1/10) : (1/15) = 3 : 2

∴ Q এর শেয়ার = {(2/5) × 1000} = 400 টাকা

Question: A man runs opposite to a train at 10 km/hr. The relative speed between them is 40 km/hr. If it takes 18 seconds for the train to pass the man when he is at rest, find the length of the train.

Solution:
Relative speed = Train’s speed + Man’s speed
⇒ vt + 10 = 40 
⇒  vt = 40 - 10
⇒  vt = 30 × (5/18) = 25/3 m/s
∴ vt = 25/3 m/s
 
∴ Length of the train = Speed × Time to cross
= (25/3) × 18
= 150 meters

Question: The ratio of speed of a motor-boat to that of the current of water is 5 : 1. The boat goes along with the current in 3 hours. It will come back in-

Solution:
Since the ratio 5 : 1 is given.
Let the speed of boat in still water = 5 km/hr and speed of stream = 1 km/hr
Now,
Downstream speed = 5 + 1 = 6 km/hr
Upstream speed = 5 - 1 = 4 km/hr

∴ Distance = Downstream speed × downstream time = 6 × 3 = 18km

∴ Upstream time = Distance/upstream speed = 18/4 = 4.5 hours

Question: A woman complete a journey in 8 hours. She travels first half of the journey at the rate of 30 km/hr and second half at the rate of 20 km/hr. Find the total journey in km.

Solution:
Let, Total distance = x
⇒ {(1/2)x/30} + {(1/2)x}/20 = 8
⇒ (x/30) + (x/20) = 16
⇒ (2x + 3x)/60 = 16
⇒ 5x = 16 × 60
⇒ x = (16 × 60)/5
∴ x = 192

So, the total distance is 192 km.

Quesation: A tank is filled in 8 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Solution:
Suppose,
pipe A alone takes x hours to fill the tank. Then, pipes B and C will take (x/2) and (x/4) hours respectively to fill the tank.

Now,
⇒ (1/x) + (2/x) + (4/x) = 1/8 
⇒ 7/x = 1/8
⇒ x = 56
∴ Pipe A alone takes 56 hours to fill the tank.

Question: Two trains, one from Dhaka to Chattogram and the other from Chattogram to Dhaka, start simultaneously. After they meet, the first train takes 4 hours to reach Chattogram, and the second train takes 9 hours to reach Dhaka. What is the ratio of their speeds?

Solution:
Time taken by Train₁ = 4 hours, And Train₂ = 9 hours.

∴ Ratio of speeds (Train₁ : Train₂) =  Speed of Train₁/Speed of Train₂
= √(9/4) = 3/2 = 3 : 2

Question:  A person can swim in water with a speed of 14 km/hr in still water. If the speed of the stream is 4 km/hr, what will be the time taken by the person to go 90 km downstream?

Solution:
Speed downstream = Speed of Boat in still water + Speed od the stream =
(14 + 4)km/hr.
= 18 km/hr.

∴ Time taken to travel 90 km downstream = Distance/Speed
= (90/18)hrs
= 5 hrs.

Question: X can do 1/5 of a work in 8 days, Y can do 25% of the work in 50 days and Z can do 1/4 of the work in 15 days. Who will complete the work first?


Solution:
X এর সম্পূর্ণ কাজ করতে সময় লাগবে = (5 × 8) = 40 দিন

Y এর সম্পূর্ণ কাজ করতে সময় লাগবে = {50 × (100/25)} = 200 দিন

Z এর সম্পূর্ণ কাজ করতে সময় লাগবে = (4 × 15) = 60 দিন

∴ X পুরো কাজ সবচেয়ে কম সময়ে, অর্থাৎ 40 দিনে শেষ করতে পারবে।