**What will you learn in this chapter?**

- Concepts and formulae based on work and time

**Previous article**

Please read Basic Concepts of Time and Work before starting this article.

**Concepts and Formulae based on Time, Work and Workforce**

So from the relationships discussed in the previous article, we can mathematically derive these three rules:

- Work is directly proportional to Person. That is, if there is more work, more people are required.Â
- Time is inversely proportional to Person. That is, if there is more people, less time is required.
- Work is directly proportional to Time. That is, if there is more work, more time is required.

Now, let us look at some formulae based on these concepts.

- Suppose â€˜Mâ€™ number of people can do a work in â€˜Tâ€™ time. Now, we know that work is directly proportional to both person and time. So, the total work can be given as:

Total Work = M Ã— TÂ

- If Ravi can do a given work in â€˜Dâ€™ number of days, then his 1-day work is given as 1D.

If 1D is the work done by Ravi in 1 day, then part of his work completed in â€˜nâ€™ days is given as 1DÃ—n. This is also termed as the Unitary Method.

- If â€˜Mâ€™ denotes the number of men, â€˜Dâ€™ denotes the number of days, â€˜Hâ€™ denotes the number of hours per day and â€˜Wâ€™ denotes the amount of work, then the relation among the four can be given as:

(MÃ—DÃ—H)/W=Constant

Where â€˜constantâ€™ can be a numeric value of some sort.

- If â€˜M1â€™ can do â€˜W1â€™ work in â€˜D1â€™ days where he/she works â€˜H1â€™ hours per day and if â€˜M2â€™ can do â€˜W2â€™ work in â€˜D2â€™ days where he/she works â€˜H2â€™ hours per day, then the relation between the two can be given as:

(M1Ã—D1Ã—H1)/W1= (M2Ã—D2Ã—H2)/W2

- If Ravi is a good worker than Suresh by â€˜xâ€™ times, then

- Ratio of work done by Ravi and Suresh is given as x:1
- Ratio of time taken by Ravi and Suresh to complete a given task is given as 1:x. This means that Ravi will take (1/x)th of the time taken by Suresh to do a particular work.Â

- If Ravi and Suresh can complete a given work in â€˜pâ€™ and â€˜qâ€™ days respectively, then together they will complete the task in pq/(p+q) days. When they are working together, they will complete (p+q)/pqth part of the given work in 1 day.Â
- If Ravi, Suresh and Rajesh can do a piece of work in â€˜pâ€™, â€˜qâ€™ and â€˜râ€™ days respectively, then together they can finish the work in pqr/(pq+qr+rp) days.
- If Ravi can do a piece of work in â€˜pâ€™ days and Ravi and Suresh together can do the same work in â€˜qâ€™ days, then number of days required by Suresh alone to do the same work is pq/(p-q) days.Â
- If Ravi and Suresh can do a piece of work in â€˜pâ€™ days, Suresh and Rajesh can do it in â€˜qâ€™ days and Rajesh and Ravi can do it in â€˜râ€™ days, then:

- Days required when Ravi, Suresh and Rajesh work together is given by 2pqr/(pq+qr+rp).
- Ravi alone can do the work in 2pqr/(pq+qr-rp).
- Suresh alone can do the work in 2pqr/(-pq+qr+rp).
- Rajesh alone can do the work in 2pqr/(pq-qr+rp).

- Suppose Ravi and Suresh can complete the work in â€˜dâ€™ days. If Ravi working alone takes â€˜pâ€™ days more than he and Suresh working together and Suresh working alone takes â€˜qâ€™ days more than he and Ravi working together, then d= sqrt of pq
- If a given group of men â€˜m1â€™ and women â€˜w1â€™ can do a given work in â€˜Dâ€™ days, then another group of men â€˜m2â€™ and women â€˜w2â€™ will take Dm1w1/(m2w1+m1w2) days.
- If a given group of men â€˜mâ€™, women â€˜wâ€™ and boy â€˜bâ€™ can complete a work in â€˜Dâ€™ days, then 1 man, 1 woman and 1 boy can do the work in Dmwb/(mw+wb+bm) days.
- If the number of people to do a work in changed in the ratio a:b, then the time required will be changed in the inverse ratio i.e. b:a.
- If Ravi, Suresh and Rajesh can complete a work in â€˜pâ€™, â€˜qâ€™ and â€˜râ€™ days respectively, then the ratio in which the earnings will be shared is given as 1/p:1/q:1/r=qr:rp:pq.
- Similarly, if Ravi did â€˜w1â€™ amount of work, Suresh did â€˜w2â€™ amount of work and Rajesh did â€˜w3â€™ amount of work, then the ratio in which the earnings will be shared is given as 1/w1:1/w2:1/w3=w2w3:w3w1:w1w2.
- â€˜Mâ€™ number of people can do a work in â€˜Dâ€™ days. If there were â€˜mâ€™ people more, then the work can be done in â€˜dâ€™ less days. Here, the value of â€˜Mâ€™ is given as M= m(D-d)/d.
- â€˜Mâ€™ number of people can do a work in â€˜Dâ€™ days. If there were â€˜mâ€™ people less, then the work can be done in â€˜dâ€™ more days. Here, the value of â€˜Mâ€™ is given as M= m(D+d)/d.
- Ravi takes â€˜pâ€™ days to do a work and Suresh takes â€˜qâ€™ days to do the same work. Both of them started working together, however after â€˜aâ€™ number of days, Ravi left. So, the total number of days required to complete the work is given as =q(a+p)/(p+q) .Â