## Time and Work

 If a person can do a work in x days, then person's one day work = 1 x

 If a person one day work is 1 , then person complete the work in n days. n

Number of days required to completed the whole work
 = whole work work done in one day

#### 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 )

 Efficiency ∝ 1 Time

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, work done by P in one day = 1 x

 work done by Q in one day = 1 y

 Then work done in one day when P and Q work together = 1 + 1 x y

 = ( x + y ) xy

Hence, number of days required to complete the whole work

 = whole work work done in one day

 = 1 { ( x + y ) / xy }

 ∴ Time taken by ( P + Q ) to complete the work = xy ( x + y)

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?

 Solution:- Work done by A in one day = 1 20

 Work done by B in one day = 1 25

 Work done in one day, when A and B work together = 1 + 1 20 25

 = ( 5 + 4 ) = 9 100 100

 Hence, required number of days = 1 = 100 9 / 100 9

 = 11 1 days 9

By formula,
Here, x = 20 and y = 25

 Numbers of days = xy ( x + y )

 = ( 20 × 25 ) = 500 ( 20 + 25 ) 45

 = 100 = 11 1 days 9 9

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

 Here, Work done by ( P + Q )'s in one day = 1 x

 and work done by P's in one day = 1 y

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

 = 1 - 1 x y

 = ( y - x ) xy

 ∴ Time taken by Q to complete the work alone = 1 { ( y - x )/ xy }

 ∴ Time taken by Q to complete the work alone = xy ( y - x )

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

 ∴ Time taken by B = xy ( y - x )

 = 10 × 14 ( 14 - 10 )

 = 140 = 35 days 4

#### 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, Work done by P in one day = 1 x

 Work done by Q in one day = 1 y

 Work done by R in one day = 1 z

 Then work done in one day when all of them working together = 1 + 1 + 1 x y z

 = ( yx + xz + xy ) xyz

Hence, number of days required to complete the whole work

 = whole work work done in one day

 Hence, number of days required to complete the whole work = 1 ( yx + xz + xy ) / xyz

 ∴ Time taken by ( P + Q + R ) to complete the work = xyz ( xy + yx + zx )

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

 Hence, required number of days = xyz ( xy + yx + zx )

 = (10 × 20 × 30) ( 10 × 20 + 20 × 30 + 30 × 10 )

 = 6000 (200 + 600 + 300)

 = 6000 = 60 1100 11

 = 5 5 days 11

#### 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

 Hence, Work done by ( P + Q )'s in one day = 1 .......( 1 ) x

 Work done by ( Q + R )'s in one day = 1 .......( 2 ) y

 Work done by ( R + P )'s in one day = 1 .......( 3 ) z

Adding Eq. (1), (2) and (3), we get

 P + Q + Q + R + R + P = 1 + 1 + 1 x y z

 or, 2 ( P + Q + R )'s one day's work = ( xy + yz + zx ) xyz

 ∴ ( P + Q + R )'s one day's work = ( xy + yz + zx ) xyz

 Hence, number of days required to complete the whole work = 1 { xy + yz + zx ) / 2xyz }

 ∴ P, Q and R can do that work in = 2xyz ( xy + yz + zx )

 P alone will finish the work in = 2xyz ( xy + yz - zx )

 Q alone will finish the work in = 2xyz ( yz + zx - xy )

 R alone will finish the work in = 2xyz ( zx + xy - yz )

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

 Required number of days = 2xyz ( xy + yz + zx )

 = 2 × 12 × 15 × 20 ( 12 × 15 + 15 × 20 + 20 × 12 )

 = 7200 ( 180 + 300 + 240 )

 = 7200 = 10 days 720

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

 M1D1T1 = M2D2T2 W1 W2

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 = ?

 M1D1T1 = M2D2T2 W1 W2

15 × 24 × 12 = 27 × x × 10

 x = 15 × 24 × 12 × 10 = 16 days 27

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 = ?

 M1D1T1 = M2D2T2 W1 W2

22 × 16 = 32 × x

 x = 22 × 16 = 11 days 32

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

 Total time = ( x + T ) y ( x + y )

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.

 Solution:-B's one day work = 1 30

 B's 10 days' work = 10 = 1 30 3

 Remaining work = 1 - 1 = 2 3 3

 ( A + B )'s one day's work = 1 + 1 20 30

 = 3 + 2 = 5 60 60

 ( A + B )'s one day's work = 1 12

 ( A + B ) finish 1 work in one day. 12

 ( A + B ) will finish 2/3 work in = 2 ÷ 1 3 12

 = 2 × 12 = 8 days 3

∴ Total time = 10 + 8 = 18 days
By Formula,
Here, x = 20, y = 30 and T = 10

 ∴ Total time = ( x + T ) y ( x + y )

 = ( 20 + 10 ) 30 ( 20 + 30 )

 = 30 × 30 = 18 days 50

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

 Total time = ( y + T ) x ( x + y )

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.

 Solution:-A's one day's work = 1 15

 A's 7 days' work = 7 = 15

 Remaining work = 1 - 7 = 8 15 15

 ( A + B )'s one day work = 1 + 1 15 25

 = 5 + 3 = 8 75 75

 ( A + B ) finish 8 work on one day. 75

 ( A + B ) will finish 8/15 work in = 8 ÷ 8 15 75

 = 8 × 75 = 5 15 8

∴ Total time = 5 + 7 = 12 days
By Formula,
Here, x = 15, y = 25 and T = 7

 ∴ Total time = ( y + T ) x ( x + y )

 = ( 25 + 7 ) 15 ( 15 + 25 )

 = 32 × 15 = 12 days 40

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

 Time = ( y - T ) x ( x + y )

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?

 Solution:-B's one day's work = 1 30

 B's 10 days' work = 10 = 1 30 3

 Remaining work = 1 - 1 = 2 3 3

 ( A + B )'s one day work = 1 + 1 20 30

 = 3 + 2 = 5 60 60

 = 1 12

 ( A + B ) finish 1 work on one day. 12

 ( A + B ) will finish 2/3 work in = 2 ÷ 1 3 12

 = 2 × 12 = 8 days 3

By Formula,
Here, x = 20, y = 30 and T = 10

 ∴ ∴ Required time = ( y - T ) x ( x + y )

 = ( 30 - 10 ) 20 ( 20 + 30 )

 Required time = 20 × 20 = 8 days 50