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51. Asif started a work and left after working for 3 days. Then Brij was called and he finished the work in 9 days. Had Asif left the work after working for 4 days, Brij would have finished the remaining work in 6 days. In how many days can each of them, working alone, finish the whole work?
  A.  Asif: 6 days; Brij: 18 days
  B.  Asif: 5 days; Brij: 19 days
  C.  Asif: 7 days; Brij: 17 days
  D.  Asif: 4 days; Brij: 20 days
     
   
View Answer

Shortcut:
X, started a work and left after working d1 days. Then Y was called and he finished the work in m1 days. Had X left the work after working for d2 days, Y would have finished the remaining work in m2 days. Then, each of them X and Y, working alone finishe the whole work in
[
m2d1 − m1d2 / m2 − m1
]
days and
[
d2m1 − d1m2 / d2 − d1
]
days respectively.
Here, d1 = 3, m1 = 9, d2 = 4, m2 = 6
Using all these values in the shortcut, we get:
Asif days =
[
6 x 3 − 9 x 4 / 6 − 9
]
=
[
18 − 36 / − 3
]
=
[
− 18 / − 3
]
= 6
Hence, Asif will do in 6 days.
Brij days =
[
4 x 9 − 3 x 6 / 4 − 3
]
=
[
36 − 18 / 4 − 3
]
= 18
Hence, Brij will do in 18 days.

52. Azim can do a work in 20 days and Bhim can do the same work in 15 days. They work together for 4 days and then Azim goes away. In how many days will Bhim finish the work?
  A.  6 days
  B.  8 days
  C.  10 days
  D.  11 days
     
   
View Answer

Shortcut:
X can do a work in 'a' days and Y can do the same work in 'b' days. If they work together for 'm' days and X goes away, then Y finishes the work in
b −
(
1 +
b / a
)
m days
Here, a = 20, b = 15, m = 4
Using these values in the shortcut, we get:
= 15 −
(
1 +
15 / 20
)
4 = 15 −
(
1 +
3 / 4
)
4 = 15 −
(
4 + 3 / 4
)
4 = 15 − 7 = 8
Hence, Bhim will finish the work in 8 days.

53. Ajay can do
3 / 4
of a work in 15 days. In how many days can he finish
1 / 8
of the work?
  A.  1.5 days
  B.  3.5 days
  C.  2.5 days
  D.  4.5 days
     
   
View Answer

Shortcut:
If X can do m/n part of a work in 'a' days, then o/p part of the work will be done in 'b' days. We can calculate the value of 'b' from the given equation
a / m/n
=
b / o/p

Here, m = 3, n = 4, a = 15, o = 1, p = 8
Using these values in shortcut, we get:
15 / 3/4
=
b / 1/8

15 x 4 / 3
=
8b / 1

b =
5 x 4 / 8
=
5 / 2
= 2.5
Hence, Ajay will finish the work in 2.5 days.

54. 76 men working 6 hours a day can do a piece of work in 12 days. Find the number of days in which 102 men working 8 hours a day can do twice the work. Assume that 2 men of the first group do as much work in 1 hour as 3 men of the second group do in 1
1 / 2
hour.
  A.  23 days
  B.  25 days
  C.  27 days
  D.  29 days
     
   
View Answer

Shortcut:
Ratio of efficiency of persons in first group to the second group = E1 : E2 = (3 x 1.5) : 2x 1 = 4.5 : 2
Now use the formula: M1D1T1E1W2 = M2D2T2E2W1
∴ D2 =
38 x 12 x 6 x 4.5 x 2 / 57 x 8 x 2 x 1
= 27 days

55. Ajay alone would take 3 hours more to complete the job than if both Ajay and Biru would together. If Biru worked alone, he took 8
1 / 3
hours more to complete the job than Ajay and Biru worked together. What time, would they take if both Ajay and Biru worked together?
  A.  5 hours
  B.  3 hours
  C.  4 hours
  D.  6 hours
     
   
View Answer

Shortcut:
If X working alone takes 'a' days more than X and Y, and Y working alone takes 'b' days more than X and Y together then the number of days taken by X and Y working together is given by √ (ab) days
Here, a = 3, b = 8
1 / 3

Using these values in the shortcut, we get:
√(
3 x 8
1 / 3
)
=
√(
3 x
25 / 3
)
=
√(
25
)
= 5
Hence, they will take 5 days.

56. A man, a woman or a boy can do a job in 20 days, 30 days or 60 days respectively. How many boys must assist 4 men and 6 women to do the work in 2 days?
  A.  4 boys
  B.  8 boys
  C.  5 boys
  D.  6 boys
     
   
View Answer

Shortcut:
If X, Y and Z can do a job aone in a days, b days and c days respectively.
∴ alone time for X = a days
alone time for Y = b days
alone time for Z = c days
Now consider the following cases,
Case I: To find the amount of work done by X, Y and Z separately.
Using the formula,
Amount of work =
Number of days actually worked / alone time

and assuming that X, Y and Z have worked for m1 days, m2 days and m3 days respectively, then
amount of work by X =
m1 / a

amount of work by Y =
m2 / b

amount of work by Z =
m3 / c


Case II: If the job is complete, then add the amount of work done by X, Y and Z and equate it to 1.
i.e
m1 / a
+
m2 / b
+
m3 / c
= 1
if the job is half complete the following equation is obtained.
m1 / a
+
m2 / b
+
m3 / c
=
1 / 2

Let the required number of boys be y.
Now, using the above shortcut, we get:
(4 men's work for 2 days) + (6 women's work for 2 days) + (y boy's work for 2 days) = 1
or,
(
4 x 2 x
1 / 20
)
+
(
6 x 2 x
1 / 30
)
+
(
y x 2 x
1 / 60
)
= 1
8 / 20
+
12 / 30
+
2y / 60
= 1
24 + 24 + 2y / 60
= 1
48 + 2y = 60
2y = 60 − 48
2y = 12
y = 6 boys

57. A and B can do a job in 16 days and 12 days respectively, 4 days before finishing the job, A joins B, B has started the work alone. Find how many days B has worked alone?
  A.  5 days
  B.  4 days
  C.  6 days
  D.  7 days
     
   
View Answer

If B works alone for m days; A's amount of work + B's amount of work = 1
or,
4 / 16
+
(m + 4) / 12
= 1
m = 5
Hence, B worked for 5 days.

58. A can do a job in 20 days, B in 30 days and C in 60 days. If A is helped on every 3rd day by B and C, then in how many days, the job is finished?
  A.  15 days
  B.  12 days
  C.  10 days
  D.  16 days
     
   
View Answer

Since A is helped by B and C on every 3 rd day.
A works for 3 days while B and C work for 1 day.
1 / 20
x 3 +
1 / 30
x 1 +
1 / 60
x 1 [B and C help only on 3rd day]
3 / 20
+
1 / 30
+
1 / 60
=
1 / 5

∴ Total time for the job = 3 x 5 = 15 days.

59. A, B and C together can do a work in 6 days. A alone can do the work in 12 days and B alone can do the same work in 18 days. Find in what time C alone can do that work?
  A.  30 days
  B.  36 days
  C.  38 days
  D.  40 days
     
   
View Answer

Shortcut:
If X, Y and Z together can do a work in 'a' days, X alone can do the work in 'm' days and Y alone can do the work in 'n' days, then Z will do the same work in
amn / mn − a(m + n)

Here, a = 6, m = 12, n = 18
Putting these values in the shortcut, we get:
Z' days =
6 x 12 x 18 / 12 x 18 − 6(12 + 18)

=
6 x 12 x 18 / 216 − 6(30)

=
6 x 12 x 18 / 216 − 180

=
6 x 12 x 18 / 36
= 36
Hence, C will do the work in 36 days.

60. Azam can complete a work in 25 days and Billu can do the same work in 10 days. If Azam after doing 5 days, leaves the work, find in how many days Billu will do the remaining work?
  A.  4 days
  B.  6 days
  C.  9 days
  D.  8 days
     
   
View Answer

Shortcut:
If X and Y can do a work in 'a' days and 'b' days respectively. X leaves the work after woring for 'z' days, then Y will do the remaining work in
(a − z)b / a

Here, a = 25, b = 10, z = 5
Putting these values in the shortcut, we get:
Billu =
(25 − 5)10 / 25
=
20 x 10 / 25
= 8 days
Hence, Billu will do the work in 8 days.

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