**BASICS OF TIME AND WORK :**

Rule : If A can do a piece of work in D days, then A's 1 day's work = 1/D

Rule : If A's work efficiency is 2 times as compared to B then

Ratio of work done by A and B is 2 :1

Ratio of times taken by A and B to finish a work = 1 : 2 (inverse ratio)

Ex. A does a work in 10 days and B does the same work in 15 days. In how many days they together will do the same work ?

A's One day work = 1/10,

B's One day work = 1/15,

Both A and B's one day work combined = (1/10 + 1/15) = 1/6

So both will finish the work in 6 days.

Ex. If A and B together can do a job in x days and A alone can do the same job in y days, then how many days it taken by B to complete the same job ?

A and B 's one day work combined = 1/x,

A's one day work is = 1/y,

Therefore B's one day work = 1/x - 1/y = (y - x)/xy

B completes the job in xy/(y - x) days.

**Rule : Let P1 person can do W1 work in D1 days with time T1 and P2 person can do W2 work in D2 days with time T2, then the relationship can be written as :**

**P1 x D1 x T1 x W2 = P2 x D2 x T2 x W1 ____________@1**

Ex. If 40 people can make 60 toys in 8 hrs, if 8 people leave the work, how many toys can make in 12 hrs ?

Solution : here P1= 40, D1= 8 hrs, W1= 60, P2= 32, D2 =12 hrs, W2= ?

Using the Rule @1, we get

P1 x D1 x T1 x W2 = P2 x D2 x T2 x W1

40 x 8 x W2 = 32 x 12 x 60,

W2 = (32 x 12 x 60)/(40 x 8)

W2 = 72 toys,

**Rule : If A, B and C can do a work in x, y and z days respectively then all of them working together can finish the work in :**

**xyz/(xy + yz + xz) number of days,**

Ex. A and B can do a piece of work in 12 days , B and C in 15 days and C and A can do in 20 days. How long would each take separately to do the same work ?

Ans.

**2(A + B + C)'s Work = (12 x 15 x 20)/(12x15 + 12x20 + 15x20) = 5 Days**

A+B+C can do the work in 5*2= 10 days

**A's Work = (A + B + C) - (B + C)Work = (10x15)/(15-10) = 30days,**

**B's Work = (A + B + C) - (C +A)Work = (20x10)/(20-10) = 20 days,**

**C's Work = (A + B + C) -(A + B)Work = (12x10)/(12-10) = 60 days.**

Ex. If 3 men or 4 women can do a work in 43 days, how long will 7 men and 5 women take to comlete the work ?

Solution:

3 men can complete 1 / 43 of work in a day

1 man can complete 1 /(43*3) work in a day

4 women can complete 1/ 43 of work in a day

1 woman can complete 1 / (43*4) of the work in a day

**7 men and 5 women can complete (7/(43x3) + 5/(43x4)) = 1/12 of work in a day**

So, 7 men and 5 women will complete the work in 12 days

Ex. Seven men can complete a work in 12 days. They started the work and after 5 days, two men left. In how many days will the work be completed by the remaining men ?

Solution :

7 men 1 day's work will be 1 / 12

1 man 1 day work will be 1 / (12*7) = 1 / 84

7 men 5 days work will be 5 / 12, so remaining work will be (1- 5/ 12) = 7 / 12

5 mens 1 day work will be 5 / 84

**7/12 work is done by them in (84/5 x 7/12) = 49/5 days**

Ex. A man, a women and a boy can complete a job in 3, 4 and 12 days respectively. How many boys must assist 1 man and 1 woman to complete the job in 1/4 of a day ?

Solution :

1 man's + 1 woman's one day work = 1 / 3 + 1 / 4 = 7 / 12

work done by 1 man + 1 woman in 1 / 4 day = (7 / 12 )*( 1 / 4 )= 7 / 48

So the work remains=(1-7/48)= 41 / 48

Also work done by one boy in 1 / 4 day = (1 / 12 )*(1 / 4 ) = 1 / 48

Since 1 / 48 work is done by 1 boy in 1 / 4 days

So 41 / 48 work can be done by ( 41 / 48 ) * (48 / 1) = 41

Ex. 10 men and 15 women together can complete a work in 6 days. It takes 100 days for one man alone to complete the same work.How many days will be required for one woman alone to complete the same work ?

Solution :

1 man's one day work = 1 / 100

( 10 men's + 15 women's ) 1 day work= 1 / 6

15 women's one days work = (1 / 6 )- ( 10 / 100 ) = 1 / 15

1 woman 1 day work = ( 1 / 15* 15 ) = 1 / 225

So one woman will complete the work in 225 days

Ex. If 6 men and 8 boys can do a piece of work in 10 days while 26 men and 48 boys can do the same in 2 days, the time taken by 15 men and 20 boys in doing the same type of work will be :

Solution :

(6 men and 8 boys ) 1 day work = 1 / 10

This can be written as equation form, let 1 man's one day work is M and one boy's one day work is B

6 M + 8 B = 1 / 10

Similarly 15 M + 20 B = 1 / 2 on solving M = 1 / 100, B = 1 / 200

Now ( 15 men + 20 boy ) 1 day's work = ( 15 / 100 )+ ( 20 / 100 ) = 1 / 4

So 15 men and 20 boys can do the work in 4 days.