Days of work is a very interesting and exciting concept of quantitative aptitude. You must have come across such questions. In this section, however, we will deal with a subsection that we know as the problem of getting “days from work “. We will see how these questions are framed in our exams. We will also state the formulae that we will use to solve these questions. Let us begin by stating the principle of these problems.

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## Days From Work

The questions in this section are very intriguing. Suppose there is some task, some physical work that a certain number of people are set to do. Suppose three people are digging a well. They finish this task in four days. If we increase the number of workers, the time taken to complete the task will decrease. The whole task here is to find the rate at which work is done per day. For example, let us see the following solved example.

We will have to develop a formula first. First, we have that Work = Number of days (Time taken) (T or D) × Number of men (M). In other words, W = D × M.

Suppose w_{1} is the work done in the first case and w_{2} in the second case. Then the ratio is equal to w_{1}/w_{2} = (T_{1} × N_{1})/(T_{2} × N_{2})

#### Example

Two people, A and B complete a certain work in six days. Another person C, if working alone can do the same work in ten days. If both the people work together, then they can do this work in how many days?

A) 15 days B) 5 days C) 4.55 days D) 3.75 days

Answer: There may be many methods that you will use to solve this. We will discuss the best and the easiest one here. The first step is to find the rate at which the work is done. If ‘n’ is the number of days in which some work is done, then work done per day = 1/n. Let us see the given question now.

As per the first condition, when A and B work together then they complete the work in 6 days. Therefore, the rate is equal to 1/6. Similarly, when C does the work alone, the same task takes ten days. So the rate is equal to 1/10 for the case when C is alone.

The rate determines the fraction of work that happens. Therefore when all the people work together then the total rate is equal to = 1/6 + 1/10. Thus, we can write it as 4/15. You might be tempted to take the reciprocal and take the answer as 15 but the numerator should be one for that. Therefore taking the reciprocal and making the denominator equal to 1, we have:

The number of days in which all three complete the task is equal to 15/4 or 3.75 days. Hence the correct option is D) 3.75 days.

### Solved Examples For You

Example

Q. Ten men can cut eight trees in 16 days. If we have to cut ten trees and we have six men, then how long will it take these men to cut the trees?

A) 23 days B) 29 days C) 17 days D) 33.3 days

Answer: To solve this question, we will have to develop a formula first. First, we have that Work = Number of days (Time taken) (T or D) × Number of men (M). In other words, W = D × M.

Suppose w_{1} is the work done in the first case and w_{2} in the second case. Then the ratio is equal to w_{1}/w_{2} = (T_{1} × N_{1})/(T_{2} × N_{2})

Here we will find the rate at which one man does the work in one day for both the cases. Substituting the values in the above equation, we have:

8/10 = (16 × 10)/(T_{2} × 6). Therefore we have T_{2} = 33.3 days.

Therefore the correct option is D) 33.3 days.

Example

Q. A can do a job in 10 days. B can do a job in 5 days. In how many days they can complete the job if they work together?

Answer: The rate at which A can do the work is = 1/10. The rate at which B can do the work is equal to 1/5. This means, in one day A and B together can do 1/10 + 1/5 of work.

Therefore, Number of days A and B together take to do 100% of work =50/15 or 3.33 days.

## Practice Problems

Q 1: Khan and Samip can do a certain task in 8 days. Samip and Yawer can do the same job in 12 days. Khan, Samip, and Yawer if working together can do the job in 6 days. In how many days can Khan and Yawer complete the job?

A) 8 days B) 16 days C) 32 days D) 6 days

Ans: A) 8 days

Q 2: Two persons A and B together can do a piece of work in 8 days. A alone does the same work in 12 days. Then if B alone works, he can do the same work in?

A) 100 days B) 33.33 days C) 24 days D) 80 days

Ans: C) 24 days.

Q 3: Three people A, B, and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

A) 10 days B) 15 days C) 20 days D) 25 days

Ans: B) 15 days