# Time and Work Formula, Tricks and Concepts

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298 Time and Work Formula, Tricks and Concepts. Time and Work is a very important topic for all ages. It is required in exams and also in daily life. Here some Formula, Tricks and Concepts of time and work are given below.

## Time and Work: Definition

Time: The number in hours, days, etc. which is needed for the completion of the task.

### Assumptions:

1. We will take is total work as 1 unit.
2. Total work done by someone is uniform throughout the whole working time.
1. The person’s T- day work = 1
2. The one day work of the person = 1/T

### Time and Work: Basic Formulae

Here are some basic formula of time and work.

#### Formula 1

If N1 people work T1 a day and they complete W1 units of work in D1 days, and N2 people work T2 a day and they complete W2 units of work in D2 days then, (N1.T1.D1)/W1 = (N2.T2.D2)/W2

1. If work is W1 and W2 is not mentioned, it can be said that (N1.T1.D1) = (N2.T2.D2)
2. If the working hours are not mentioned drop it.

#### Formula 2

If A can do a work in x days. B is k times more efficient than A. B can do the work in x/k days. Both can together do it in x/(k+1) days.

#### Formula 3

If one person can finish a work in x days and another person can finish the same work in y days respectively, then together they can finish the work in [(x.y)/(x+y)] days.

#### Formula 4

If two people can do work together in x days, but one person can do the same work in y days, then the other person can do it in [(x.y)/(xy)] days.

#### Formula 5

Three people A, B and C can finish a work separately in t1, t2 and t3 days respectively and the work is finished after A work on it for d1 days, B for d2 days and C for d3 days.

Then, d1/t1 + d2/t2 + d3/t3 = 1

#### Formula 6

If A and B are together and they can do a piece of work in x days, B and C together can do it in y days and C and A together can do it in z days, then number of days required to do the same work:

1. A, B, and C are working together = 2xyz/(xy+yz+zx)
2. If A is working alone = 2xyz/(xy+yz−zx)
3. B is working alone = 2xyz/(−xy+yz+zx)
4. If C is working alone = 2xyz/(xy−yz+zx)

### Time and Work: Other Important Formulae

• If the number of men to do a job is changed in the ratio a:b, then the time required to do the work will be changed in the inverse ratio. ie; b:a.
• If people work for the same number of days, the ratio in which the total money earned has to be shared is the ratio of work completed per day by each one of them.

A, B, C can do a piece of work in x, y, z days respectively. The ratio in which the amount earned should be shared is 1/x:1/y:1/z=yz:zx:xy

## Time and Work: Problems

Let us do some questions.

### Question 1

A can do a piece of work in 60 days, which B can do in 40 days. Both started work. Now A left 10 days before the completion of the work. The work was finished in how many days?

Ans:

A left the job 10 days before the completion.

B worked alone for the last 10 days.

Now calculate B’s 10 days of work, which he did alone.
In 10 days B will do 10 × 1/40 = 1/4th of the work.
Remaining work 1 – ¼ = ¾ (A and B have done together).

A and B can do 1/60 + 1/40 work in 1 day.

Therefore, one day work is

1/60 + 1/40 = (2 + 3)/120 = 5/120 = 1/24.

They can finish the work in 24 days.
They would have done 3/4th of the work in 24 × 3/4 = 18 days.
⇒ Total days = 18 + 10 = 28 days.

### Question 2

A did a piece of work in 24 days and B in 20 days but with the help of C, they finished the work in 8 days. C alone can do the work in how much time?

Ans: We use the above formulas

(1/A) + (1/B) + (1/C) = (1/8), now
(1/C) = (1/8) – (1/A) – (1/B)

⇒ (1/C) = (1/8) -(1/24) = (1/20)

⇒ (1/C) = (1/30)
Thus, C can do this work in 30 days.

### Question 3

A can do as much work in 6 days as C in 10 days. B can do as much work in 6 days as C can do in 4 days. What time would B require to do a work if A takes 48 days to finish it?

Sol.

A : C = 6 : 10 or (A/C) = (3/5)

B : C = 6 : 4 or (B/C) = (3/2)

(B/A) =(B/C) x (C/A) = (3/2) x (5/3) = (5/2)
Thus, B = (5/2) x A = (5/2) x 48 = 120 days.

### Question 4

A can do a work in 15 days and B in 20 days. If they do the same work together for 4 days, then the fraction of the work that is left :

Ans.

1 day’s work of A = 1/15
1 day’s work of B= 1/20
(A + B)’s 1 day work = (1/15 + 1/20) = 7/60
(A + B)’s 4 day work = (7/60 * 4) = 7/15
Therefore, the work that is remaining = (1 – 7/15) = 8/15

### Question 5

Vee can do a work in 3 days while Ray can do the same work in 2 days. Both of them finishes work together and get Rs. 150. What is the share of Vee?

Vee’s wages : Ray’s wages

= Vee’s 1 day work : Ray’s 1 day work

= 1/3 : 1/2 = 2:3
Vee’s share = 2/5 * 150 = Rs. 60

### Question 6

We see that 12 Men or 18 women can reap a field in 14 days. Find the number of days that 8 men and 16 women will take to reap it?

12 men = 18 women

or 1 man = 3/2 women
Thus 8 men + 16 women = (8 × 3/2 + 16) women = 28 women
Now 18 women can reap the field in 14 days
Therefore, 28 women can reap it in = 9 days

### Question 7

A is thrice as efficient as B and is, therefore, able to finish a piece of work 10 days earlier than B. In how many days A and B will finish it together?

WC = 3:1
WT = 1:3
x —->3x
1/x – 1/3x = 1/10
x = 20/3
3/20 + 1/20 = 1/5 => 5 days

### Question 8

A work which could be finished in 9 days was finished 3 days earlier after 10 more men joined. The number of men employed was?

x —–> 9
(x + 10) —-> 6
Therefore, x * 9 = (x + 10)6
x = 20

### Question 9

Tina can do a certain job in 12 hours. Working independently, Ann can do the same job in 9 hours. If Tina Works independently at the job for 8 hours and then Ann works independently, how many hours will it take Ann to complete the remainder of the jobs?

Work done by Tina in 1 hr = 1/12
Work done by Tina in 8 hrs = 1/12 x 8 which is 2/3

The remaining work is 1/3 which is completed by Ann in 1/39 which is 3 hrs.

### Question 10

A can do a piece of work in 40 days; B can do the same in 30 days. A started alone but left the work after 10 days, then B worked at it for 10 days. C finished the remaining work in 10 days. C alone can do the whole work in?