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Working together, Jose and Jane can complete an assigned task in 20 days. However, if Jose worked alone and complete half the work and then Jane takes over the task and completes the second half of the task, the task will be completed in 45 days. How long will Jose take to complete the task if he worked alone? Assume that Jane is more efficient than Jose.
Explanatory Answer
Let us assume that Jose will take x days to complete the task if he works alone and that Jane will take y days to complete the task if she worked alone.
From the information provided in the first statement of the question, we know that they will complete the task in 20 days, if they worked together on the task.
Therefore,1/x + 1/y = 1/20
From the second statement, we can conclude that x/2 + y/2 = 45
Or, x + y = 90 => x = 90 - y.
Substituting the value of x as 90 - y in the first equation, we get 1/(90-x) + 1/y = 1/20
Or y2 - 90 + 1800 = 0.
Factorizing and solving for y, we get y = 60 or y = 30.
If y = 60, then x = 90 - y = 90 - 60 = 30 and
If y = 30, then x = 90 - y = 90 - 30 = 60.
As the question clearly states that Jane is more efficient than Jose, the second answer is the only possible alternative.
Hence, Jose will take 60 days to complete the task if he worked alone and Jane will take only 30 days to complete the same task.

A can complete a project in 20 days and B can complete the same project in 30 days. If A and B start working on the project together and A quits 10 days before the project is completed, in how many days will the project be completed? Explanatory Answer If A can do complete a project in 20 days, then A will complete 1/20th of the project in a day
. Similarly, B will complete 1/30th of the project in a day.
Let the total number of days taken to complete the project be x days.
Then, B would have worked for all x days, while A would have worked for (x - 10) days.
Therefore, A would have completed (x-10)/20th of the project and B would have complete x/30 th of the project.
i.e., (x-10)/20 + x/30 = 1
Solving for x, we get x = 18.

Ram, who is half as efficient as Krish, will take 24 days to complete a work if he worked alone. If Ram and Krish worked together, how long will they take to complete the work? the correct answer is 8 days.
Explanatory Answer: Ram takes 24 days to complete the work, if he works alone.
As Krish is twice as efficient as Ram is, Krish will take half the time to complete the work when Krish works alone, i.e., in 12 days.
Ram completes 1/24 th of the work in a day.
Krish completes 1/12 th of the work in a day.
When they work together, they will complete 1/24 + 1/12 = 1/8 th work in a day.
Therefore, when they work together they will complete the work in 8 days.

Explanation for Question 4: One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall? Answer-12/7
One woman and one man can build a wall together in two hours,
One woman and two girls can build a wall together in two hours
from the above two we can say that: efficiency of one man = Efficiency of two girls.
given that one man and one girl can do a work in four hours.
it means 3 girls can finish the work in 4 hrs. i.e 1 girl can finish the workin 12hrs.
1 man can finish the work in 6 hrs.
1 women can finish the work in 3 hrs.Assume total work be 12 units. 1 girl will do one unit per hour. 1 man will do 2 units per hour. 1 women will do 4 units per hour.
Therefore,work will be finished in 12/7 hours

Explanation for Question 5:Machine A and Machine B can produce 1 widget in 3 hours working together at their respective constant rates. If Machine A’s speed were doubled, the two machines could produce 1 widget in 2 hours working together at their respective rates. How many hours does it currently take Machine A to produce 1 widget on its own? Solution: A+B requires 3 hours to produce 1 widget.
3(A+B) requires 1 hour to produce 1 widget.
2A+B requires 2hrs to produce 1 widget.
2(2A+B) requires 1hr to produce 1 widget
therefore 3(A+B)= 4A+2B
A=B,i.e efficiency of A and B are same. A can finish 1 widget in 6hrs.

Explanation for Question 6:A can do a job 12 days, B is 60% more efficient than A. How many days B will take to complete the job alone ?
Answer:7.5
Solution:
Assume "A" does 10 units per day. then total work is 10X12= 120 unit
As "B" is 60% more efficient "B" does 16 units per day.
To complete 120 units of work "B" requires 120/16 days= 7.5.

Explanation for Question 7: 7.Ram can finish a job in 18 days. Shaam can do the same job within 9 days. If they work togother what portoin of work they can complete in a day?
Solution:1/18+1/9=1/6.

Explanation for Question8:Adam can do a job in 15 days, Eve can do the same job in 20 days. If they work together for 4 days on this job. What fraction of job is incomplete ?
Solution:Assume total work be 60 units(lcm of 15 and 20).
Adam will do 60 units in 15 days.
Therefore he will do 4 units per day.
Eve can do 60 units in 20 days
Therefore Eve does 3 units per day.
together they can do 7 unts per day
in four days they finish 4x7=28 units
Leftover work is (60-28)=32units.
Fraction of work left over is 32/60=8/15.

Explanation for Question9:Ram takes 6 hours to type 32 pages. Shaam takes 5 hours to type 40 pages. In how many hours they will type 110 pages together ?
Solution:Ram will type 32/6 pages in 1hr.
Shaam will type 40/5 pages in 1hr
Together they will type (32/6+40/5) pages in 1 hr.
There dore to type 110 pages they will take 110/(32/6+40/5)hr=8hr 15min.

Explanation for Question 10: 7 men take 12 days to complete a job. They worked for 7 days, after 7 days 2 men left the job. In how many days 5 men will complete the job ?
Answer is 7.
Solution:
7 men take 12 days to complete a job
But 7 days two men left.
If there were 7 members they can finsish the remaing work in 5 days
Therefore 5 members can finish the work in 7 days (m₁d₁=m₂d₂)

Explanation for Question 11:.A can do 1/4 of the job in 3 days. B can do 1/6 of the same job in 4 days. If they work together $ 180 is to be paid to them in total, how much A should get ?
Answer – 120
Solution:A can do 1/4 of the job in 3 days
it means A can finish the work in 12 days.
B can do 1/6 of the same job in 4 days
it means B can finish the work in 24 days
Therefore A is twice as efficient as B.
The amount will be shared in the ratio 2:1.
A's share =(2/3)of 180=120.

Explanation for Question 12: If A, B and C can do a job in 20, 30 and 60 days respectively. In how many days A can do the work if B and C help him on every third day ?
Answer:
SOLUTION: Let us assume total work be 60 uinits (lcm of 20, 30, 60).
A can do 60 units of work in 20 days.
A can do 3units/day.
B can do 60 units of work in 30 days
B can do 2units/day.
C can do 60 units of work in 60 days
C can do 1unit/day.
A,B and C together will do (3+2+1) units/day. i.e 6 units/day.
A works for two days and on third day all three together will work.
In three days the work done will be (3+3+6)=12units/cycle.
1 cycle = 3 days
To do 60 units of work the time taken will be (60 units)/( 12units/cycle ) = 5 cycles = 5 X 3days= 15 days.

Explanation for Question 13:.5 men and 2 boys working together can do four times as much work as a man and a boy. Working capacities of a man and a boy are in the ratio:
Explanatory Answer
Efficiency of (5m+2b)= 4 times the efficiency of ( 1m+1b)
Efficiency of (5m+2b)= efficiency of ( 4m+4b)
Efficiency of 1m= efficiency of 2b
Efficiency of 1m= 2 times efficiency of 1b
(Efficiency of 1m/ efficiency of 1b)=2/1.

Explanation for Question 14:If 12 men and 16 boys can do a piece of work in 5 days; 13 men and 24 boys can do it in 4 days, then the ratio of the daily work done by a man to that of a boy is:
Explanatory Answer
5 (12m+16b)= 4(13m+24b)
on simplification m/b=2/1.

Explanation for Question 15Sixteen men can complete a work in twelve days. Twenty-four children can complete the same work in eighteen days. Twelve men and eight children started working and after eight days three more children joined them. How many days will they now take to complete the remaining work?
Explanatory Answer
efficiency of 16X12men= Efficiency of 24X18 children
efficiency ratio of men to children is (24X18)/(16X12)= 9/4.
4 men = 9 children=> 12men=27 children.
12 men + 8 children= 35 children.
Assume each children will do 1 unit per day, then total work=24x18.
Twelve men and eight children worked for eight days= 35 X 8
After 8 days work left = 24X18-35X8=152
As 3 more children joined the group the no. of children will be 38.
152/38=4 days

Explanation for Question 16:10 women can complete a work in 7 days and 10 children take 14 days to complete the work. How many days will 5 women and 10 children take to complete the work?
Explanatory Answer
10 women can complete a work in 7 days
10 children take 14 days to complete the work.
i.e from the above two we can say that two children can be replaced by 1 women.
5 women and 10 children efficiency is same as 10 women.
There for the answer is 7 days.

Explanation for Question 17:Twelve children take sixteen days to complete a work which can be completed by eight adults in twelve days. Sixteen adults started working and after three days ten adults left and four children joined them. How many days will they take to complete the remaining work? Explanatory Answer
Efficiency of 16X12 children = Efficiency of 8X12 adults
efficiency ratio of adult to children=2/1.
Assume each children will do one unit per day.
Total work =16X12=192.
16 adults can be replaced with 32 children.
In three days the work done=32X3=96.
Total no. of children after three days(10 adults left and 4 children joined)=16.
Remaining (192-96=96) units of work done in 96/16=6days.

Explanation for Question 18: Twelve children take sixteen days to complete a work which can be completed by eight adults in twelve days.

Column A Column B

Efficiency of 3 children Efficiency of 2 adults

Solution:Efficiency of 16X12 children = Efficency of 12X 8 adults.
Efficiency ratio of adult to children=2/1.
From this we can say that efficency of 2 adults is greater than the efficiency of 3 children.
Column B is greater.

Explanation for Question 19:12 men can complete a piece of work in 4 days, while 15 women can complete the same work in 4 days. 6 men start working on the job and after working for 2 days, all of them stopped working. How many women should be put on the job to complete the remaining work, if it is to be completed in 3 days?
Solution:12 men can complete 100% of work in 4 days.
6 men can complete 25% of work in two days
15 women can complete 100% of work in 4 days
It means in one day 15 women can do 25% of the work.
Threfore in three days 15 women can complete 75% of the work.
The Answer is 15.

Explanation for Question 20:A can do a piece of work in 10 days; B in 15 days. They work for 5 days. The rest of the work was finished by C in 2 days. If they get $ 1500 for the whole work, the daily wages of B and C are:
Solution:A can earn $1500 if he alone works for 10 days.
Per day A will get $150
B can earn $1500 if he alone works for 15 days.
Per day B will get $100
For the five days of work A & B together will get 5(150+100)= $1250
The remaining amount($250) will go to C for two days of work
Per day C will get $125
Daily B and C will get (100+125)= $225.

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Column A	Column B
Efficiency of 3 children	Efficiency of 2 adults