Concept of Efficiency :-
Efficiency, means rate of doing work. This means that more the efficiency, less will be the number of days required to complete a certain work and less the efficiency, more will be the number of days required to complete a certain work.
Efficiency is inversely proportional to time ( work is constant )
If A is twice as efficient as B, then
A does twice as much work as B in the same time interval.
A will require half the time as required by B to do the same work.
Work Done :-
Consider a whole work as the unit work.
1. Work done by two persons :-
Let P can do a whole work in x days and Q can do the same one unit work in y days.
Hence, number of days required to complete the whole work
Ex- If A can do a work in 20 days and B can do the same work in 25 days, then how many days will they take to complete the work both while working together?
By formula, Here, x = 20 and y = 25
2. When P and Q can do a work in x days and P alone can finish that work in y days, then work completed by Q
Work done by Q's in one day = Work done by ( P + Q )'s in one day - work done by P's in one day
Ex- Mohan and sohan can do a work in 10 days and mohan alone can do it in 14 days. In how many days can sohan alone do it?
Solution:- Here, x = 10 and y = 14
3.Work done by three persons :-
Let P can do a whole work in x days, Q in y days and R in z days.
Hence, number of days required to complete the whole work
Ex- If A, B and C can do a work in 10, 20 and 30 days respectively, then how many days will they take to complete the work when all the three work together.
Solution:- Here, x = 10, y = 20 and z = 30
4. Work done by together :-
Let P and Q can do a whole work in x days, Q and R can do the same work in y days and R and P can do it in z days, then work completed by P, Q and R together
Adding Eq. (1), (2) and (3), we get
Ex- A and B can do a work in 12 days. B and C can do the same work in 15 days, while C and A can do it in 20 days. Find the time in which A, B and C can finish the work, working together.
Solution:- Here, x = 12, y = 15 and z = 20
5. If M1 persons can do W1 work in D1 days T1 hours per day and M2 persons can do W2 work in D2 days T2 hours per day, then
Ex- If 15 men working 12 h per day can reap a field in 24 days, in how many days can 27 men reap the field working 10 h per day ?
Solution :- Let the work completed in x days.
Given, M1 = 15, D1 = 24, T1 = 12, M2 = 27, T2 = 10, D2 = ?
15 × 24 × 12 = 27 × x × 10
Ex- 22 men can complete a job in 16 days. In how many days, will 32 men complete that job?
Solution:- Let the work completed in x days.
Given, M1 = 22, D1 = 16, M2 = 32, D2 = ?
22 × 16 = 32 × x
6. If A can do a piece of work in x days and B can do the same work in y days . Both begin together, if A leaves the work T days before its completion, then total time taken for completion of work will be given as
Ex- A can do a piece of work in 20 days while B can do it in 30 days. They begin together but 10 days before the completion of the work, A leaves off. Find the total number of days for the work to be completed.
∴ Total time = 10 + 8 = 18 days
By Formula,
Here, x = 20, y = 30 and T = 10
7. If A can do a piece of work in x days and B can do the same work in y days . Both begin together, if B leaves the work T days before its completion, then total time taken for completion of work will be given as
Ex- A can do a piece of work in 15 days while B can do it in 25 days. They begin together but 7 days before the completion of the work, B leaves off. Find the total number of days for the work to be completed.
∴ Total time = 5 + 7 = 12 days
By Formula, Here, x = 15, y = 25 and T = 7
8. If a can do a piece of work in x days and B can do the same work in y days. Both begin together but after some days. A leaves off and the remaining work is completed by B in T days. Then time after which A left
Ex- A and B can do a piece of work in 20 days and 30 days, respectively. Both begin together but after a certain time, A leaves off. In this case B finishes the remaining work in 10 days. After how many days did A leave?
By Formula, Here, x = 20, y = 30 and T = 10