ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: An outlet pipe can empty a cistern in 4 hours. In what time will it empty 3/4 part of the cistern?

Solution:
The outlet pipe empties one complete cistern in 4 hours
Time taken to empty 3/4 Part of the cistern = (3/4) × 4 = 3 hours.

Question: A pipe can fill a tank in m hours and another pipe can empty it in n (n > m) hours. If both pipes are open, in how many hours will the tank is filled?

Solution:
Net part filled in 1 hour = (1/m - 1/n)
= (n - m)/mn hours.

∴ The tank will be filled in = mn/(n - m) hours.

Question: Three pipes A, B and C can fill a tank in 6 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 6 hours. The number of hours taken by C alone to fill the tank is

Solution: 
A, B and C together can fill in one hour = 1/6
in two hours = 2/6 = 1/3

the remaining part is = 1 - 1/3 = 2/3

in 6 hours, A and B can do = 2/3
so, in one hour A and B can do = 2/(3 × 6) = 2/18

so, C can do in one hour = (A, B and C in one hour) - (A and B in one hour)
= 1/6 - 2/18
= 1/18

hence, C take 18 hours to fill the tank

Question: A pipe can fill a cistern in 16 hours. After half the tank is filled, three more similar taps are opened. What is the total time to fill the cistern completely?

Solution:
Time is taken to fill half of the tank = (1/2) × 16 = 8 hrs
In 1 hour pipe can fill = 1/16 part filled by 4 pipes in1 hour = 4 × (1/16) = 1/4 part
So, remaining half part = 4 × (1/2) = 2 hours
∴ Total time = 8 + 2 = 10 hours.

Question: A swimming pool is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is?

Solution:
Suppose, the first pipe alone takes x hours to fill the tank.
Then, the second and third pipes will take (x - 5) and (x - 9) hours respectively to fill the tank.

ATQ,
1/x + 1/(x - 5) = 1/(x - 9)
⇒ (x - 5 + x)/x(x - 5) = 1/(x - 9)
⇒ (2x - 5)/(x² - 5x) = 1/(x - 9)
⇒ 2x² - 18x - 5x + 45 = x² - 5x
⇒ 2x² - 23x + 45 - x² + 5x = 0
⇒ x² - 18x + 45 = 0
⇒ x² - 15x - 3x + 45 = 0
⇒ x(x - 15) - 3(x - 15) = 0
⇒ (x - 15)(x - 3) = 0
∴ x = 15 [neglecting x = 3]

So, first pipe alone takes 15 hours to fill the tank.

Question: A cistern can be filled by a tap in 6 hours while it can be emptied by another tap in 9 hours. If both the taps are opened simultaneously, then after how much time the cistern will get filled?

Solution:
The cistern fill in 1 hour = ( 1/6 ) - ( 1/9 ) part = 1/18 part
The cistern fill 1/18 part = 1 hour 
The cistern fill full = ( 1 × 18 ) /1 hour = 18 hours

Question: A pump can fill a tank with water in 2 hours. Because of a leak, it took 140 minutes to fill the tank. The leak can drain all the water of the tank in-

Solution: 
একটি পাইপ চৌবাচ্চা পূর্ণ করতে পারে ২ ঘণ্টায় বা ১২০ মিনিটে  
১ মিনিটে পূর্ণ করে ১/১২০ অংশ 

একটি ছিদ্র থাকায় তা পূর্ণ করতে পারে ১৪০ মিনিটে 
১ মিনিটে পূর্ণ হয় ১/১৪০ মিনিটে 

ছিদ্র দিয়ে ১ মিনিটে খালি হয় = (১/১২০) - (১/১৪০)
= (৭ - ৬)/৮৪০
= ১/৮৪০ অংশ 

সম্পূর্ণ অংশ খালি করতে সময় লাগে = ১/১/৮৪০ মিনিট 
= ৮৪০ মিনিটে 
= ৮৪০/৬০ ঘণ্টায় 
= ১৪ ঘণ্টায় 

Question: A tank is 30% full with water. If 18 liters of water is added the tank becomes 3/4 full. What is the capacity of the tank?

Solution:
Let, Capacity of the tank is x Liters.

ATQ,
30% of x + 18 = (3/4) × x
⇒ (30x/100) + 18 = 3x/4
⇒ (3x/10) + 18 = 3x/4
⇒ (3x/4) - (3x/10) = 18
⇒ (15x - 6x)/20 = 18
⇒ 9x = 18 × 20
⇒ 9x = 360
∴ x = 40

∴ The capacity of tank is 40 Liters.

Question: A tank can be filled by pipe A in 2 hours and pipe B in 6 hours. At 10 A.M. pipe A was opened. At what time will the tank be filled if pipe B is opened at 11 A.M.?
 
Solution: 
A ২ ঘণ্টায় পূর্ণ করে ১ অংশ 
১ ঘণ্টায় পূর্ণ করে ১/২ অংশ 

বাকি থাকে (১ - ১/২) অংশ 
= ১/২ অংশ ; যা A, B একসাথে সম্পন্ন করে। 

B ৬ ঘণ্টায় করে ১ অংশ কাজ 
১ ঘণ্টায় করে ১/৬ অংশ কাজ 

A, B ১ ঘণ্টায় করে (১/৬) + (১/২) অংশ 
= ৪/৬ অংশ
= ২/৩ অংশ

A, B ২/৩ অংশ পূর্ণ করে ১ ঘণ্টায় 
১/২ অংশ পূর্ণ করে ৩/(২ × ২)
= ৩/৪ ঘণ্টায় 

মোট সময় = ১ + (৩/৪) = ৭/৪ ঘন্টা = ১ ঘণ্টা ৪৫ মিনিট 
∴ ট্যাঙ্ক পূর্ণ হবে = ১০ টা + ১ ঘণ্টা ৪৫ মিনিট 
= ১১ টা ৪৫ মিনিট  

Question: A leak in the bottom of a tank can empty the whole tank in 8 hours. An inlet pipe fills water are the rate of 6 liters a minute. When the tank is full, the inlet is opened, and due to the leak, the tank is empty in 12 hours. How many liters does the tank hold? 

Solution:
Work done by the inlet pipe in 1 hour = (1/8 - 1/12) = (3 - 2)/24 = 1/24
Work done by the inlet pipe in 1 minute = (1/24 × 1/60) = 1/1440

 Volume of 1/1440 part = 6 liters
 Volume of the whole tank = (1440 × 6) = 8640 liters

Question: A tank can be filled by pipe A in 5 hours and emptied by pipe B in 8 hours respectively. How much time will it take for the tank to be half full?

Solution:
Pipe alone can fill the tank in = 5 hrs.
Pipe alone can empty the tank in = 8 hrs.
Let, the tank to be half full in x hrs, 

ATQ,
x/5 - x/8 = 1/2
⇒ (8x - 5x)/40 = 1/2
⇒ 3x/40 = 1/2
⇒  3x = (1/2) × 40
⇒  3x = 20
∴ x = 20/3 hours

Question: A tank is filled in 5 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:
Let,
pipe A alone takes x hours to fill the tank.
Then, pipe B take x/2 hours to fill the tank.
and pipe C will take x/4 hours to fill the tank.

ATQ,
(1/x) + (2/x) + (4/x) = 1/5
⇒ 7/x = 1/5
∴ x = 35 hours.