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21. Subhash can copy 50 pages in 10 hours; Subhash and Prakash together can copy 300 pages in 40 hours. In how much time can Prakash copy 30 pages?

Discuss
Solution:
Number of page copied by (Subhash + Prakash) in 1 hour = 30040  = 7.5 pages;
Subhash copied pages in one hour = 5010 = 5 pages
Hence, Prakash copied pages in one hour = 7.5 - 5 = 2.5
Thus,
Prakash can copied 30 pages in = 302.5 = 12 hour
22. An engineer undertakes a project to build a road 15 km long in 300 days and employs 45 men for the purpose. After 100 days, he finds 2.5 km of the road has been completed. Find the (approx.) number of extra men he must employ to finish the work in time.

Discuss
Solution:
Variation Method:
In 100 days only 2.5 km road i.e. 16.66 % of work has been completed
Men Days Road (km)
45 100↓ 2.5
x↑ 200 12.5↑

Arrows show the directions of variation of quantity with respect to each other
x45=100×12.5200×2.5
x = 113 men;
Required men to be increased,
= 113 - 45
= 68
23. There is provision of food in fort for 1200 soldiers for 60 days. After 15 days, 200 soldiers leave the fort. Remaining food will last for how many days?

Discuss
Solution:
Work equivalence method:
1200 × 45 = 1000 × x
Hence, x = 54 days

Variation Method:
After 15 days 200 soldiers leaved.
Soldiers              Food for days
1200 ↓                 45
1000                 ↑ x (let)
Arrows show the opposite variation to each other
12001000=x45
Or, x = 1200×451000   = 54 days.
24. A and B working together completed a job in 5 days. If A works twice as efficiently as he actually did and B works 13 of actual efficiency, the work would have completed in 3 days. Find the for A to complete the job alone.

Discuss
Solution:
One Day's work of A and B together,
1A+1B=15........(i)

When A works with twice efficiency,Then,
2A+13B=13........(ii)

on solving equations (i) and (ii), we get
A=254=614
25. Two pipes A and B can fill a cistern in 12 min and 16 min respectively. Both the pipes are opened together for a certain time but due to some obstruction the flow of water was restricted to 78 of full flow in pipe A and 56 of full in pipe B. This obstruction is removed after some time and tank is now filled in 3 min from that moment. How long was it before the full flow.

Discuss
Solution:
Let the obstruction remain for x min.
Hence,
Part of cistern filled in X min + part of cistern filled in 3 min = full cistern
[7x8×12+5x6×16]    + [312+316]   = 1
12x96+716=1
Thus,
X = 4.5 min.
26. Three pipes A,B and C attached to a cistern. A can fill it in 10 min, B in 15 min, C is a waste pipe for emptying it. After opening both the pipes A and B, a man leaves the cistern and returns when the cistern should have been just full. Finding, however, that the waste pipe had left open, he closes it and the cistern now gets filled in 2 min. In how much time the pipe C, if opened alone, empty the full cistern?

Discuss
Solution:
Let pipe C alone can empty the cistern in x min.
A fills cistern in 1 min = 110
B fills cistern in 1 min = 115
A and B together fill in 1 min =10×1510+15=15025=6min
Since, waste pipe was left open for 6 min then,
6 min, 6x part of cistern will be emptied by C
Now,
6x part of the cistern would be filled by A and B in 2 min.
Hence,
cistern will be filled in 3x min. And
x3 = 6
x = 18 min.
27. There is a group of 5 boys and 2 girls. The two groups working together can do four times as much work as a boy and a girl. Ratio of working capacities of a boy and a girl is:

Discuss
Solution:
Let 1 boy's 1 day's work = x
And 1 girl's 1 day's work = y
Now,
(5 boys + 2 girls)'s work = 5x + 2y
Given ,
5x + 2y is equal to 4 times work done by a boy and a girl
Thus,
5x + 2y = 4(x + y)
5x + 2y = 4x + 4y
x = 2y
xy=21
Hence, the required ratio is 2 : 1
28. A group of 12 men can do a piece of work in 14 days and other group of 12 women can do the same work in 21 days. They begin together but 3 days before the completion of work, man's group leaves off. The total number of days to complete the work is:

Discuss
Solution:
Let x be the required number of days
Given,
12 men and 12 women can complete a work separately in 14 days and 21 days respectively
Then,
12 men's 1 day work = 114
And,
12 women's 1 day work = 121
Then ,
12 women's 3 days work = 321 = 17

The remaining work = 1−17  = 67
Man's group leaves 3 days before the completion of work
That is, they were working together for x - 3 days
Thus, we have 17 work left to be done in last 3 days by the women's group. This also means 67 th of work has been done by both the groups (before men left)
Now, (12 men + 12 women)'s 1 day work = 114+121   = 542
i.e., 542 work is done by 2 groups in 1 day.
So, 67 of work is done by 2 groups together in 425×67   = 365 days
Total time take to complete the work will be
= 365 + 3 = 515
29. Vimal can do a piece of work in 20 days, Vimal and Kamal together can do in 12 days. If Kamal does the work only for half a day daily then in how many days the work will be completed ?

Discuss
Solution:
Vimal's 1 day work = 120
Since, Vimal and Kamal can together complete in 12 days
i.e. (Vimal + Kamal)'s 1 day work = 112
Then,
Kamal's 1 day work,
=112−120⇒260⇒130
If Kamal Works only for half a day daily, then his 1 day work becomes 12×130  = 160
Therefore, 1 day work of both Vimal and Kamal,
=120+160⇒460⇒115
Hence, the work will be completed in 15 days.
30. There are three boats A, B and C, working together they carry 60 people in each trip. One day an early morning A carried 50 people in few trips alone. When it stopped carrying the passengers B and C started carrying the people together. It took a total of 10 trips to carry 300 people by A, B and C. It is known that each day on an average 300 people cross the river using only one of the 3 boats A, B and C. How many trips it would take to A to carry 150 passengers alone?

Discuss
Solution:
Combined efficiency of all the three boats = 60 passengers /trip
Now, consider option (A)
15 trips and 150 passengers means efficiency of A = 10 passengers per trip
A's efficiency = 10 passengers per trip
Then, (B + C) combined efficiency = 50 passengers per trip
Since, combined efficiency is 60 so option (A) is correct

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