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31. Two pipes can separately fill a tank in 10 hours and 15 hours respectively. Both the pipes are opened to fill the tank but when the tank is
1 / 3
full a leak develops in the tank through which
1 / 3
of the water supplied by both the pipes leak out. What is the total time taken to fill the tank?
  A.  5 hours
  B.  8 hours
  C.  7 hours
  D.  10 hours
     
   
View Answer

Shortcut:
Two pipes can separtely fill a tank in 'a' hours and 'b' hours respectively. If both the pipes are opened to fill the tank but when the tank is 'n' part full a leak develps in the tank through which 'm' part of the total water supplied by both the pipes leak out, then the total time to fill the tank is
ab / a + b
x
1 − mn / 1 − m
hours.
Here, a = 10, b = 15, n =
1 / 3
, m =
1 / 3

Using these values in the shortcut, we get:
Required Time =
10 x 15 / 10 + 15
x
1 − 1/3 x 1/3 / 1 − 1/3

=
10 x 15 / 25
x
1 − 1/9 / (3 − 1)/3

= 6 x
(9 − 1)/9 / 2/3

= 6 x
8/9 / 2/3

= 6 x
8 x 3 / 2 x 9
= 6 x
4 / 3
= 2 x 4 = 8
Hence, the total time to fill the tank is 8 hours.

32. A cistern is normally filled in 4 hours but takes 1 hour longer to fill because of a leak in its bottom. If the cistern is full, the leak will empty it in:
  A.  10 hours
  B.  15 hours
  C.  20 hours
  D.  25 hours
     
   
View Answer

Shortcut:
A cistern is normally filled in 'a' hours but takes 'h' hours longer to fill because of a leak in its bottom. If the cistern is full, the leak will empty it in
a x (a + h) / h
hours.
Here, a = 4, h = 1
Using these values in the shortcut, we get:
Required Time =
4 x (4 + 1) / 1
= 4 x 5 = 20
Hence, the total time to empty the tank is 20 hours.

33. If three taps are opened together, a tank is filled in 6 hours. One of the taps can fill it in 3 hours and another in 6 hours. How does the third tap work?
  A.  2 hours
  B.  1 hours
  C.  5 hours
  D.  3 hours
     
   
View Answer

Shortcut:
If three taps are opened together, a tank is filled in 'h' hours. One of the taps can fill it in 'a' hours and another in 'b' hours. The third tap fills or empties the tank in
h / 1 − h[(a + b)/ab]
hours.
Note: The nature of the third tap - whether it is filler or waste pipe depends upon the +ve or −ve sign of the above expression.
Here, t = 6, a = 3, b = 6
Using these values in the shortcut, we get:
Required Time =
6 / 1 − 6[(3 + 6)/3 x 6]

=
6 / 1 − 6[9/18]

=
6 / 1 − 6[1/2]

=
6 / 1 − 3

=
6 / − 2
= − 3
Hence, −ve sign shows that the third pipe is a waste pipe which empties the tank in 3 hours.

34. Two pipes P and Q can separately fill in 15 and 5 minutes respectively and a waste pipe C can carry off 17 litres per minute. If all the pipes are opened when the cistern is full, it is emptied in 1 hour. How many litres does the cistern hold?
  A.  60 litres
  B.  55 litres
  C.  65 litres
  D.  50 litres
     
   
View Answer

Shortcut:
Two pipes can fill a cistern in 'a' and 'b' minutes respectively. A waste pipe carries of 'z' litres of water per minute from the cistern. If all three pipes are opened together and a full cistern gets emptied in 'm' minutes, then the capacity of the cistern is
z(abm) / ab + am + bm
litres.
Here, a = 15, b = 5, z = 17, m = 60
Using these values in the shortcut, we get:
Required capacity =
17(15 x 5 x 60) / 15 x 5 + 15 x 60 + 5 x 60

=
17 x 75 x 60 / 1275
= 60
Hence, the cistern holds 60 litres.

35. There are 5 filling pipes, each capable of filling a cistern alone in 6 minutes, and 3 emptying pipes each capable of emptying a cistern alone in 8 minutes. All pipes are opened together and as a result, tank fills 11 litres of water per minute. Find the capacity of the tank.
  A.  19 litres
  B.  27 litres
  C.  24 litres
  D.  21 litres
     
   
View Answer

Shortcut:
To find out the capacity (C) of the cistern in litres, if 'a' number of filling pipes, each capable of filling a cistern alone in 'y' minutes, and 'b' number of emptying pipes, each capable of emptying a cistern alone in 'z' minutes, are opened together and as a result 'q' is the rate at which the tank fills per minute, the following formula is used, C =
qyz / az − by
litres.
Here, a = 5, y = 6, b = 3, z = 8, q = 11
Using these values in the shortcut, we get:
Required capacity =
11 x 6 x 8 / 5 x 8 − 3 x 6

=
11 x 6 x 8 / 40 − 18

=
11 x 6 x 8 / 22
= 3 x 8 = 24
Hence, the capacity of the tank is 24 litres.

36. Two pipes can fill a cistern in 7 and 8 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom,
4 / 15
hours extra are taken for the cistern to be filled up. If the cistern is full, in what time would the leak empty it?
  A.  52 hours
  B.  56 hours
  C.  50 hours
  D.  62 hours
     
   
View Answer

Shortcut:
Two pipes X and Y can fill a cistern in 'a' and 'b' hours respectively. If opened together they take 'z' hours extra to fill the cistern due to a leak, then the time in which the leak alone empties the full cistern is
ab / a + b
x
[
1 +
ab / (a + b)z
]
hours.
Here, a = 7, b = 8, z =
4 / 15

Using these values in the shortcut, we get:
Required time =
7 x 8 / 7 + 8
x
[
1 +
7 x 8 / (7 + 8)4/15
]

=
7 x 8 / 15
x
[
1 +
7 x 8 / 15 x 4/15
]

=
7 x 8 / 15
x
[
1 +
7 x 8 / 4
]

=
7 x 8 / 15
x [1 + (7 x 2)]
=
7 x 8 / 15
x 15 = 7 x 8 = 56
Hence, the leak will empty the cistern in 56 hours.

37. A cistern has a leak which would empty it in 8 hours. A tap is turned on which admits 3 litres a minute into the cistern, and it is now emptied in 12 hours. How many litres does the cistern hold?
  A.  3320 litres
  B.  4420 litres
  C.  4220 litres
  D.  4320 litres
     
   
View Answer

Shortcut:
A cistern has a leak which would empty it in 'a' hours. If a tap is turned on which admits water at the rate of 'z' litres per hour into the cistern, and the cistern is now emptied in 'b' hours, then the capacity of the cistern is z x
ab / b − a
litres.
Here, a = 8, b = 12, z = 3 x 60 = 180
Using these values in the shortcut, we get:
Required quantity = 180 x
8 x 12 / 12 − 8

= 180 x
8 x 12 / 4
= 180 x 2 x 12 = 4320
Hence, the cistern holds 4320 litres.

38. One filling pipe P is 8 times faster than second filling pipe Q. If Q can fill a cistern in 36 minutes, then find the time when the cistern will be full if both fill pipes are opened together.
  A.  7 minutes
  B.  2 minutes
  C.  4 minutes
  D.  9 minutes
     
   
View Answer

Shortcut:
One filling pipe X is 'z' times faster than the other filling pipe Y. If Y can fill a cistern in 'a' hours, then the time in which the cistern will be full, if both the filling pipes are opened together, is
a / z + 1
hours.
Note: Value of the slower filling pipe is given.
Here, a = 36, z = 8
Using these values in the shortcut, we get:
Required time =
36 / 8 + 1
=
36 / 9
= 4
Hence, the cistern will get full in 4 minutes.

39. One filling pipe A is 8 times faster than second filling pipe B. If A can fill a cistern in 36 minutes, then find the time when the cistern will be full if both fill pipes are opened together.
  A.  32 minutes
  B.  35 minutes
  C.  25 minutes
  D.  30 minutes
     
   
View Answer

Shortcut:
One filling pipe X is 'z' times faster than the other filling pipe Y. If Y can fill a cistern in 'a' hours, then the time in which the cistern will be full, if both the filling pipes are opened together, is z x
a / z + 1
hours.
Note: Value of the faster filling pipe is given.
Here, a = 36, z = 8
Using these values in the shortcut, we get:
Required time = 8 x
36 / 8 + 1

= 8 x
36 / 9
= 8 x 4 = 32
Hence, the cistern will get full in 32 minutes.

40. One fill pipe A is 5 times faster than second fill pipe B and takes 72 minutes less than the fill pipe B. When will the cistern be full if both fill pipes are opened together?
  A.  11 minutes
  B.  15 minutes
  C.  14 minutes
  D.  9 minutes
     
   
View Answer

Shortcut:
If one filling pipe X is 'z' times faster and takes 'a' minutes less time than the other filling pipe Y, then the time, they will take to fill a cistern, if both the pipes are opened together, is
za / z2 − 1
minutes. X will the cistern in
a / z − 1
minutes and Y will take to fill the cistern
za / z − 1

Note: Here, X is the faster filling pipe and Y is the slower one.
Here, z = 5, a = 72
Using these values in the shortcut, we get:
Required time =
5 x 72 / (5)2 − 1

=
5 x 72 / 25 − 1
=
5 x 72 / 24
= 5 x 3 = 15
Hence, the cistern will get full in 15 minutes.

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