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51. A is thrice good a workman as B and therefore is able to finish a job in 40 days less than B. Working together they can do it in:

Discuss
Solution:
Given,
A is thrice good workman as B. Means,
A = 3B
Let B can finish work in X days, then A will finish same work in (X - 40) days alone
Now,
BX = 3B × (X - 40)
X = 60 days
B can finish work in 60 days, then A can finish the work in 20 days.
One day work of B = 160
One day work of A = 120
One day work of (A+B) =
160+120⇒1+360⇒115
So, they can finish work together in 15 days
52. A and B can compete a piece of work in 18 days. They worked together for 12 days and then A left. B alone finished the work in 15 days. If Rs. 1500 be paid for the work then A's share is:

Discuss
Solution:
A and B can complete the work in 18 days, work rate = 10018 = 5.55% per day
They together can complete the work in 12 days = 5.55 × 12 = 66.60%
Now, A leaves and B takes another 15 days to complete the whole work, Work rate of B = 33.3015  = 2.22% per day
B work for (12 + 15) = 27 days
So, Work done by B in 27 days = 2.22 × 27 ≈ 60% And So 40% work is done by A. so there share should be 60% and 40% ratio.
A's share = 40% of 1500 = Rs. 600
53. If 3 men or 4 women can plough a field in 43 days, how long will 7 men and 5 women take to plough it?

Discuss
Solution:
3 men or 4 women can plough the field in 43 days
3 men = 4 women
1 man = 43 women
7 man = 283 women
7 men and 5 women = 5+283  = 433 women
4 women can plough field in 43 days
So,
1 women can plough in = 43 × 4 days
Hence,
433 women can plough = 43×4×343   = 12 days

54. To complete a piece of work A and B take 8 days, B and C 12 days. A, B and C take 6 days. A and C will take :

Discuss
Solution:
Given (A+B)'s one day's work = 18
(B + C)'s one day's work = 112
(A + B + C) 's 1 day's work = 16

Work done by A, alone= (A + B + C) 's 1 day's work - (B + C)'s one day's work
=16−112=2−112=112
Work done by C, alone = (A + B + C) 's 1 day's work - (A + B)'s one day’s work
=16−18=4−324=124
⇒ (A + C)’s one day’s work
=112+124=2+124=324=18
⇒ (A + C) will take 8 days to complete the work together
55. Two pipes can fill the cistern in 10hr and 12 hr respectively, while the third empty it in 20hr. If all pipes are opened simultaneously, then the cistern will be filled in:

Discuss
Solution:

Work done by all the tanks working together in 1 hour,
⇒110+112−120=215
Hence, tank will be filled in 152 = 7.5 hour.

56. Three taps A, B and C together can fill an empty cistern in 10 minutes. The tap A alone can fill it in 30 minutes and the tap B alone in 40 minutes. How long will the tap C alone take to fill it?

Discuss
Solution:
A, B and C together can fill 100% empty tank in 10 minutes
Work rate of (A + B + C) = 10010 = 10% per minute
A alone can fill the tank in 30 minutes
Work rate of A = 10030 = 3.33% per minute
B alone can fill the tank in 40 minutes
Work rate of B = 10040 = 2.5%
Work rate of (A + B) = 3.33 + 2.5 = 5.83% per minute
Work rate of C,
= Work rate of (A + B + C) - (A + B)
= 10 - 5.83 = 4.17% per minute
So, C takes = 1004.17 ≈ 24 minutes to fill the tank
57. A and B working separately can do a piece of work in 9 and 15 days respectively. If they work for a day alternately, with A beginning, then the work will be completed in:

Discuss
Solution:
Work rate of A = 1009 = 11.11% work per day
Work rate of B = 10015 = 6.66% work per day

They together can do (A + B) = 11.11 + 6.66 ≈ 18% work per day
They are working in alternate day, so we take 2 days = 1 unit of day
Therefore, in one unit of day they can complete 18% of work
(A + B) can complete 90% of work in 5 units of days. i.e. (5 × 18)
And the rest 10% work will be completed by A in Next day
So, total number of day,
= 5 Unit of days + 1 day of A
= 2 × 5 + 1 = 11 days
58. Two pipes A and B can fill a tank in 36 min. and 45 min. respectively. Another pipe C can empty the tank in 30 min. First A and B are opened. After 7 minutes, C is also opened. The tank filled up in:

Discuss
Solution:
Pipe A can fill empty tank in 36 min.
Pipe A can fill the tank = 10036 = 2.77% per minute
Pipe B can fill empty tank in 45 min.
Pipe B can fill the tank = 10045 = 2.22% per min.
A and B can together fill the tank
= (2.77 + 2.22) ≈ 5% per minute
So, A and B can fill the tank in 7 min.
= 7 × 5 = 35% of the tank
Rest tank to be filled = 100 - 35 = 65%
C can empty the full tank in 30 min.
C can empty the tank = 10030 = 3.33% per min.
C is doing negative work i.e. emptying the tank
A, B and C can together fill the tank,
= 2.77% + 2.22% - 3.33% = 1.67% tank per minute
So, A, B and C will take time to fill 65% empty tank,
= 651.67 = 39 min. (Approx)
59. Three men A, B, C working together can do a job in 6 hours less time than A alone, in one hour less time than B alone and in one half the time needed by C when working alone. Then A and B together can do the job in:

Discuss
Solution:
Time taken by A =x hours.
Therefore taken by A, B and C together = (x - 6)
Time taken by B = (x - 5)
Time taken by C = 2(x - 6)
Now, rate of work of A + Rate of work of B + Rate of work of C = Rate of work of ABC.

⇒1x+1x−5+12(x−6)=1x−6
On solving above equation, we get x = 3, 406
When x = 3, the expression (x - 6) becomes negative, thus it's not possible.
⇒x=406
Time taken by A & B together = 1320+35
= 43 hours
60. A does half as much work as B in one -sixth of the time.If together they take 10 days to complete a work, how much time shall B take to do it alone?

Discuss
Solution:
To do half of the work in one sixth of the time means A is 3 times as efficient as B. To understand this point just think if same work would have been done in one sixth of the time then A would have been six times as efficient as B.
Now they together complete the work in 10 days.
B is working with someone who is 3 times as efficient as himself.
That means 4 people of B's efficiency finished the work in 10 days.
So B alone would have done it in 40 days.
A alone would have taken one third of this time.

Alternatively,
Given,
A×16=B×12
So, A = 3B
Now,
Given they together complete the work in 10 days
So, One Day''s work of,
(A + B) = 10010 = 10%
(3B + B) = 10%
4B = 10%
So, one day work of B = 104 = 2.5%
So, B can complete 100% work in = 1002.5 = 40 days

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