ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: A tap can fill a tank in 10 hrs. After half the tank is filled, four more similar taps are opened. What is the total time taken to fill the tank completely?

Solution:
Time taken by one tap to fill half the tank = 5 hrs
Remaining part after 5 hrs = 1 - (1/2) = 1/2

total taps = 1 + 4 = 5
So, part filled by the five taps in 1 hours = 5 × (1/10) = 1/2

∴ Total time = 5 +1 = 6 hrs

Question: Three pipes A, B and C can fill a tank in 8 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 9 hours. The number of hours taken by C alone to fill the tank is?

Solution:
Part filled in 2 hours = 2/8 = 1/4
Remaining part = 1 - (1/4) = 3/4

(A + B)'s 9 hour's work = 3/4
(A + B)'s 1 hour's work = 3/36 = 1/12

∴ C's 1 hour's work = {(A + B + C)'s 1 hour's work } - {(A + B)'s 1 hour's work }
= (1/8) - (1/12)
= (3 - 2)/24
= 1/24
∴ C alone can fill the tank in 24 hours.

Question: Two pipes P and Q together can fill a cistern in 4 hours. Had they been opened separately, then Q would have taken 6 hours more than P to fill the cistern. How much time will be taken by P to fill the cistern separately?

Solution:
Let the cistern be filled by pipe P alone in x hours.
Then, pipe Q will fill it in (x + 6) hours.

ATQ,
⇒ (1/x) + {1/(x + 6)} = 1/4
⇒ (x + 6 + x)/{x(x + 6)} = 1/4
⇒ (2x + 6)/x(x + 6) = 1/4
⇒ x² - 2x - 24 = 0
⇒ (x - 6)(x + 4) = 0
∴ x = 6     ; [neglecting the negative value of x]

∴ Pipe P alone can fill the cistern in 6 hours.

Question: A booster pump can be used to fill as to empty the tank. The capacity of the tank is 1200 m³. The emptying capacity of the tank is 10 m³ per minute higher than its filling capacity and the pump requires 4 minutes lesser to vacant the tank than it requires to fill it. Calculate the filling capacity of the pump is-

Solution:
Let, the filling capacity of the pump = x m³/min

Given that,
Capacity of the tank = 1200 m³
Emptying rate is 10 m³/min more than filling rate
Emptying time is 4 minutes less than filling time

Then,
Filling time = 1200/x​ minutes
Emptying capacity = x+10 m³/min
Emptying time = 1200/(x + 10)

ATQ,
(1200/x​) - {1200/(x + 10)} = 4
(1/x) - {1/(x + 10)} = 1/300
(x + 10 - x)​/x(x + 10) = 1/300
x² + 10x - 3000 = 0
(x + 60) (x - 50) = 0

So, possible values is,
x = 50 And x = - 60 [not valid]

So, the filling capacity of the pump is 50 m³/min

 Question: It takes two pipes X and Y, running together, to fill a tank in 6 minutes. It takes A 5 minutes less than Y to fill the tank, then what will be the time taken by Y alone to fill the tank?

Solution:
Let the time taken by pipe X to fill the tank be a minutes
Time is taken by pipe Y to fill the tank = a + 5 minutes

So,
⇒ (1/a) + {1/(a + 5)} = 1/6
⇒ (2a + 5)/a(a + 5) = 1/6
⇒ a² + 5a - 12a - 30 = 0
⇒ a² - 7a - 30 = 0
⇒ (a - 10)(a + 3) = 0
⇒ a = 10, - 3
∴ a = 10  ; [neglecting the negative value of a]

Thus, time taken by Y alone to fill the tank is 10 + 5 = 15 minutes

Question: A cistern can be filled by two taps A and B in 12 hours and 16 hours respectively. The full cistern can be emptied by a third tap C in 8 hours. If all the taps are turned on at the same time, in how much time will the empty cistern be filled completely?

Solution:
A’s 1 hour’s work = 1/12
B’s 1 hour’s work = 1/16
C’s 1 hour’s work = 1/8

Therefore, (A + B + C)’s 1 hours net work= (1/12) + (1/16) - (1/8)
= (4 + 3 - 6)/48
= 1/48

So, time taken by (A + B + C) to fill the cistern = 48 hours.

Question: A water tank is two-fifth full.Pipe A can fill a tank in 6 minutes and pipe B can empty it in 10 minutes.If both the pipes are open,how long will it take to empty or fill the tank completely?

Solution:
ট্যাংকটি ২/৫ পূর্ণ
পাইপ A,
৬ মিনিটে ভর্তি করতে পারে ১ অংশ
∴ ১ মিনিটে ভর্তি করে ১/৬​ অংশ
আবার,
পাইপ B,
১০ মিনিটে খালি করতে পারে ১ অংশ
∴ ১ মিনিটে খালি করতে পারে ১/১০ অংশ

একসাথে কাজ করার হার = (১/৬​) - (১/১০) = ২/৩০ = ১/১৫ অংশ
অর্থাৎ, একসাথে কাজ করলে প্রতি মিনিটে ট্যাংকের ১/১৫ অংশ ভর্তি হয়।

∴  ট্যাংকটি বাকি থাকে = ১ - (২/৫) = ৩/৫ অংশ

∴ ৩/৫ অংশ ভর্তি হতে সময় লাগে = (১/১৫) × (৫/৩) = ১/৯ মিনিট

∴ ১ বা সম্পূর্ণ অংশ ভর্তি হতে সময় লাগে = ৯ মিনিট

Question: A tank that was 10% full was emptied into a 50-liter bucket. If the oil now fills 40% of the bucket's volume. Then what is half of the capacity of the tank in liters?

Solution:
দেওয়া আছে,
বালতির আয়তন = 50 লিটার
∴ বালতির 40% = 50 × (40/100) = 20 লিটার
তার মানে বালতিতে ট্যাংক থেকে 20 লিটার তেল ঢালা হয়।
তাহলে, ট্যাংকের 10% ধারন ক্ষমতা = 20 লিটার

∴ অর্ধেক বা 50% ধারন ক্ষমতা = 50 × (20/10) = 100 লিটার

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

Solution:
Net part filled in 1 hour

∴ Net rate = Filling rate - Emptying rate
= (1/a) - (1/b)
= (b - a)/ab

∴ The tank will be filled in = ab/(b - a) hours.

Question: Two pipes A and B can fill a cistern in 36 and 54 minutes respectively. Both pipes are opened together, after how many minutes should B be turned off, so that the cistern be fill in 24 minutes?

Solution:
A can fill the cistern in 36 minutes
So in 1 min A can fill the cistern = 1/36 th part
In 24 min, A can fill the cistern = 24/36 = 2/3 rd

∴ Remaining part = 1 - (2/3) = 1/3 rd

As B can fill full cistern in 54 minutes
So it will fill 1/3 rd part in = (1/3) × 54) = 18 minutes.