Time & Work

Note -: In conventional Method work is always treated as 1

Example: So if I say that a person can complete a work in 15 days that means he will do 1/15 work in one
 day, It's simple maths.

Now another person does the same work in 30  days. So he will do 1/30 work in 1 day.

Now if i ask in how many days they will complete the work together. What we gonna do is Add their 1 day
 of work

 like 1/15 + 1/30 = (2+1)/30 = 3/30 = 1/10 

Now this 1/10 we got is actually their 1 day work, So if they do 1/10 work in one day then it's simple they
 will complete the whole work in 10 days.

Now that was the conventional method and I believe that you all know how to solve Questions through Conventiona
l method.

So now lets move on to the Faster method i.e efficiency method.

In efficiency method the Work is not treated in numerical value, Like in Conventional method the work is 1 but here 
the work is treated as percentage.

So by common sense the work is always treated as 100%

So when i say a person completes a work in 15 days it means he will do 100/15 % work in 1 day i.e 6.66% work in
 1 day

If another person does the work in 30 days that means he will do 3.33% work in 1 day.

And together they will do 6.66 + 3.33 = 9.99 or 10% work in one day So in how many days they will do the complete
 work,
 that will be 100/10 = 10 days.

Now it may sound difficult That we have to convert Each value in % but don't worry you don't have to convert each
 value, You just have to learn all the values till 1/30 and then it will be a cakewalk.

Now we will take Some standard Cases Of time and work and you all are free to ask any problem if you have in 
any case.

Case 1 - A does a work in X days, B does a Work in Y days In how many days they will complete the work.

Question- A completes the work in 10 days and B completes the work in 15 days In how many days they
 will complete
the work.

Conventional Method 

Work done by A in 1 day = 1/10

Work Done by B in 1 day = 1/15

Work done By A & B together in 1 day = 1/10 + 1/15 = (3+2)/30 = 5/30 = 1/6

As A & B Completes 1/6 work in one day So they will complete the whole work in 6 Days.

Case -2 A can do a work in X days and B can do it Y days, In how many days the work is completed if they
 work alternatively 
Started by A.

Now in these type of question the person are not actually working together, what happens in this type of question
 is that A works for 1
 day and then on 2nd day B work and on 3rd again A work and on Fourth again B works and so on till the work is
completed.

For example A can do a work in 10 days B can do it 15 days and how many days they will finish it if The work is started
 by A

So again work done by A in one day = 1/10

 ''     ''          ''        ''      ''  B      ''      ''      = 1/15

Now the work done by Togther will be = 1/10 + 1/15 = 1/6 [ Note now this 1/6 work is not done by them in 1 day but in
 2 days Actually, See
 A worked for 1 day and did 1/10 work on the second day B worked and finished the 1/15 work So in total 2 days they
 did 1/6 work]

So in 2 days they did 1/6 work so in how many days they will complete the whole work, Simple 12 days.

Case 3: A can do a work in X days, B can do the work Y days and A leaves after working Z days.

Question- A can do a work in 10 days and B can do it in 15 days, A works for 2 days and then leaves, In 
how many days will the work be completed?

Now here we can se that A leaves after 2 days that means A and B only worked for 2 days and the remaining work is done by B alone.

So first we have to calculated the work done by A and B together in 2 days.

So work done be A in 1 day = 1/10

""   "         "        "  B "  "   "     = 1/15

Work done by A & B together is 1 day = 1/10 + 1/15 = 1/6

Work done by A & B together in 2 days = (1/6) * 2 = 1/3

So remaining work = 1 - 1/3 = 2/3

Now this 2/3 work has to be done by B alone.

So it takes 15 days for B to do Complete work, How much time it will be taken by B to do 2/3 work ? So it will be
 15*(2/3) = 10 days

So the work will be completed in 2 + 10 days = 12 days

Case 4

A can do a piece of Work in 10 days and B can do it in 15 days, In how many days will the work be
completed if B leaves 2 days before the completion on work.

Now in this question B leaves before 2 days

Together they can do the work in what = 1/10 + 1/15 = 1/6

That means 6 Days.

That means Together they could have completed the work in 6 days but B works only till 4th day and The remaining
 work will be done by A alone

So they worked together for 4 days in all So work done by them in 4 days = (1/6)*4 = 4/6 = 2/3

remaining work = 1/3 

Now this 1/3 work will be done by A alone 

Now A can do the complete work in 10 days, So the time taken by him to do 1/3 work = 10 * (1/3) = 10/3 days or
 3.33 days

So total time taken = 4+ 3.33 days = 7.33 days

Case 5: A can do a Work in X days B can Do it in Y days, In how many days The work will get completed if 
B leaves 2 days before the actual completion of work.

Question: A can do a work in 10 days B can do it in 15 days, In how many days The work will get completed
 if B leaves 2 days before the Actual Completion of Work.what is the difference between this Actual 
completion of work and Completion of Work?

See in last example the work was supposed to get completed in 6 days, So we just Solved the question taking into
 consideration that B leaves 2 days before the completion of work i.e B worked for 4 days and the rest 2 days work
 was don by A alone and Completes that work in 3.33 days Total 7.33 days.

So if i ask In this question If B left 2 days before the actual completion then it means B should have left on 5.33
 days Got it ?

Now back to the question.

Now just imagine that the work gets completed in x days.

So A would work for x days[ A works for the whole time ]

And B would work for x-2 days[ because B left 2 days before the actual completion of work]

So now According to Question

x/10 + (x-2)/15 = 1 [ Beacuse work is always one ]

(3x+2x-4)/30 = 1

5x -4 = 30

5x = 34

x = 6.8 days.

So the work will be completed in 6.8 Days.

It can also be asked That after how many days B left, So the answer would be Simple 6.8 - 2 = 4.8 days

Efficiency Method

A's Efficiency = 10%

B's Efficiency = 6.66%

Let the no. of days be x 

so Accordring to question

10x + 6.66(x-2) = 100[ Work is always 100% in efficiency method ]

10x + 6.66x - 13.33 = 100

16.66x = 113.33

x = 113.33/16.66 = 6.8 

Answer = 6.8 days