Time and Work

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/n^th 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 = 1/9^th of the total work

In the same way, the amount of work done by B in 1 day = 1/6^th 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
• 5/18^th 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

Adding all of them

• (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 1/3^rd 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?

Answers

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