# Time and Work

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These problems mainly deal with the number of workers and the time in which they can complete a given amount of work. The following principles need to be kept in mind while solving such problems.

• If a person can do a given amount of work in n days, the amount of work he does in 1 day is 1/nth of the total work.
• If the amount of work he does in one day is 1/n, the number of days he will take to complete the total work is n days.

The following rules determine how the quantities of time, work and number of men vary with respect to each other.

• If we fix the number of men, the work done is proportional to the time.
• If we fix the work, the time is inversely proportional to the number of men.

This means that if there are more men, fewer days are required.

If there are a less number of men, more days are required.

From the above two rules ,we get the following formula:

where men1, time1, and work1 are the quantities in the first case and men2, time2 and work2 are the quantities in the second case.

A few solved examples will help in elucidating these rules.

1. If A can do some work in 9 days and B can do the same work in 6 days, in how many days can both of them do it together?

The work is done by A in 9 days

The amount of work done by A in 1 day = 1/9th of the total work

In the same way, the amount of work done by B in 1 day = 1/6th of the total work

Amount of work done by A and B in one day = A’s one day work + B’s one day work

• 1/9 + 1/6
• 5/18th of the total work
• Therefore, number of days required to complete the job are 18/5 = 3.6 days

2. X and Y can do a given amount of work in 12 days. Y and Z can do it in 15 days. Z and X can do it in 20 days. Find the number of days in which X can do it on his own.

The amount of work which X and Y do in one day = 1/12

The amount of work which Y and Z do in one day = 1/15

The amount of work which Z and X do in one day = 1/20

We can also write it this way-

( X+Y)’s 1 day work = 1/12

(Y+Z)’s i day work = 1/15

(Z+X)’s 1 day work = 1/20

• (X+Y+Y+Z+Z+X)’s 1 day work = 1/12 + 1/15 + 1/20
• OR 2( X+Y+Z)’s1 day work = 1/12 + 1/15 + 1/20 = (5 + 4 + 3) / 60 = 12/60 = 1/5
• ( X+Y+Z)’s1 day work = 1/10
• X+Y+Z can do it in ten days

X’s one day work = (X+Y+Z)’s one day work – (Y+Z)’s one day work

• 1/10-1/15= 1/30
• Therefore, X can do it in 30 days

3. 2 men and 3 women can do a piece of work in 10 days. 3 men and 2 women can do it in 8 days. In how many days will 2 men and 1 woman do it?

If we assume:

1 man’s one day work to be x

1 woman’s one day work to be y

Then

• 2x + 3y = 1/10
• 3x + 2y = 1/ 8

We can solve the equation to get x= 7/200 and y= 1/100

2 men and 1 woman’s one day work = 2 * ( 7/200) + 1/100 = 8/100

• 2/25

Therefore, the number of days which they will take to finish the work = 25/2 or 12.5

4. If 6 men work for nine hours a day, they will be able to finish the work in 100 days, in how many days will 10 men finish the work if they work for six hours a day?

Using,

The time taken in the first case= number of days * number of hours = 9 * 100

We get (6 * 900)/ work1= (10 * 6* days) / work2

Since the work done in both the cases is the same work1= work2

• 6 * 900= 6 * 10 * days
• Days= 90
• It will take 90 days to finish the work.

Solve the following exercise

1. A, B, C, D can finish a piece of work in 6, 12, 20, 30 days respectively. If they work together, in how many days will they finish the work?
2. A and B finish some work in 12 days. A, B and C finish it in 8 days. In how many days will C finish it if he works alone?
3. A takes twice as much time as B to finish a given amount of work. C takes one third the time which A takes. Together they can finish the work in 2 days. In how many days can B alone do the work?
4. A completes 1/3rd of the work in 4 days. B completes the remaining work in 16 days. If A and B work together, in how many days will they finish the work ?
5. There are two boys Jim and Peter. Jim can finish a given amount of work in six days if he works for seven hours every day. Peter can finish the work in seven days, and he works for nine hours every day. How many days will both of them take to finish the work if they work for 8 2/5 hours every day?
6. Alex can do a piece of work in 12 days. Brian is 60% more efficient than Alex. In how many days will Brian finish the work on his own?
7. 100 men work for 35 days at a certain piece of work. Then 100 more men join in. In this way the number of days taken to complete the total work is 40. In how many days would the work have been completed if the extra men had not joined in?
8. 6 workers of company A and 8 workers of company B can do the same work in 81 days. In how many days can 12 workers of company A and 11 workers of B do the same work?
9. 24 men complete a given piece of work in 16 days. 32 women take 24 days to complete it. 16 men and 16 women worked on it for 12days. Now if we want to finish the work in two more days, how many men should be added?
10. A can do a piece of work in 10 days, B can do it in 12 days and C can do it in 15 days. All begin together, a leaves the work 2 days after the work begins and B leaves the work three days before the work finishes. How long did the work last?

1. 3 days
2. 24 days
3. 6 days
4. 8 days
5. 3 days
6. 7 ½ days
7. 45 days
8. 24 days
9. 24 men
10. 7 days

1. Yes, you are correct the answer to the eighth question is 24 days. Thanks for pointing that out. Here are the Solutions to Questions 9 and 10. Hope they help.
9
24 men’s one day work = 1/16
One man’s one day work = 1/ (24*16)
32 women’s one day work = 1/24
One woman’s one day work = 1/ (32 * 24)
If 16 men and 16 women work together, their one day work = 16 /(24*16) + 16/ ( 32 *24)
 1/16
 Total number of days taken by them to complete the work = 16 days

But they work only for 12 days so they do 3/4th of the work i.e. 1/4th is left

Now 16 men and 16 women’s work in 2 days = 2 * ( 1/16) = 1/8
So 1/4th of the work has to be done in two days out of which 1/8th is done by the existing workers
 The amount of work left = ¼-1/8 = 1/8

Work to be done = 1/8
Number of days = 2
So the amount of work which needs to be done each of the two days = ½ * (1/8) = 1/16
Now let the number of men who can do 1/8th of the work in 2 days = Number of men who can do 1/16th of the work in one day = x
 Number of men * each man’s one day work = 1/16th of the total work
 x/ (24 * 16 ) =1/16
 X = 24
 i.e, 24 more men are required for the last two days.

10

A’s one day work = 1/10
B’s one day work = 1/12
C’s one day work = 1/15

For the first two days all three work
Work done = number of days * one day’s work
2 ( 1/10 + 1/12 + 1/15) = ½

For the last three days only C works
Work done = number of days * one day’s work
3(1/15) = 1/5

So the total amount of work done in these 5 days ( 2+3) = ½ + 1/5 = 7/10th
The amount of work left = 1-7/10 = 3/10
Now for x days B and C work and they complete 3/10th of the total work in these x days

Work done = number of days * one day’s work
3/10 = x * (1/12 + 1/15)
3/10 = (9/60) x
X = 2
Therefore only B and C work for two days

The total number of days thus required are =
Days when A, B and C work = 2
Days when B and C work = 3
Days when C works = 2
Therefore the total number of days required = 7

2. @NLV : This is how you do the 3rd one : Let B take x number of days to finish the work. Since A takes twice as much as B, A finishes the work in 2x days and since C takes 1/3rd the time taken by A, C will finish the work in 2x/3 days! So now, as you already know, A’s one day work is 1/2x, B’s is 1/x and C’s is 3/2x, if you add all of them and equate it to 1/2, it implies, work done by A, B and C put together in one day! So if you futher simplify, you’ll get the value of x as 1/8 and its reciprocal is the number of days taken by B ( since we have assumed B to have have taken x days to complete the work) 🙂

3. 3. let the time taken by b= n
then time taken by a= 2n
and time taken by c= 2n/3 (1/3rd of a)
work done by all 3 in one day= 1/n + 1/2n + 3/2n
=6/2n = 3/n
days taken 2 complete da work = n/3
acc to que days takn by all 2 complete da wrk = 2
so, n/3 =2 or n=6
now time taken by c = 2*6 /3 = 4 days…
but ur ans is 6… how cum?????
@ neelasha: dint get da 1/8 thing…cud u plz xplain it in detail…(wid da solution)

• for 6th,
ratio of times by a and b is 160:100=8:5
b takes alone be x,
then 8:5::12:x
8x=12*5
x=15/2=7.5=7 1/2

• That solution should be wrong because if Alex takes 8 days, Brian should take 3.2 as he is 60% more efficient than Alex. 8:5 would denote that Alex is 60% less efficient than Brian and that changes the meaning of the question. Please correct me if I am missing something.

4. isn’t the 6th one supposed to be 4.8 days, because if he’s 60% more efficient then he should definitely take less than half the time to complete the work,i.e. he should take 40% of the time right?

• No, the answer should be 7.5 days, or 7 and a half days. Explanation: Brian is 60% more efficient than Alex, which means the ratio of time taken by Alex and Brain to complete a particular work should be 160:100= 8:5.
Now, Alex can do a work in 12 days; however, the number of days taken by Brian to finish the work is not known.
=> Ratio of time taken by Alex and Brian::Ratio of days taken by them
=> 8:5::12:x
=> 8x = 60
=> x = 60/8
=> x = 7.5 days
Therefore, Brian takes 7.5 days to finish the work.

5. hey,can u explain me that 2nd solved question becoz if we add work of all the three persons then we get 1/5 n ur answer is 1/10 don’t know why………..

6. (5) Jim,s 1h work is 1/42 and Peter,s 1h work is 1/63
Both 1h work is 1/42 + 1/63 = 5/126
So both will take 126/5 hours to complete the work
No of days required when both work together is
126/5 ÷ 42/5 = 3 days