What will you learn in this chapter?
  • Concepts and formulae based on work and time
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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:

  1. Work is directly proportional to Person. That is, if there is more work, more people are required. 
  2. Time is inversely proportional to Person. That is, if there is more people, less time is required.
  3. 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 ‘M’ number of people can do a work in ‘T’ 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 × T 

  • If Ravi can do a given work in ‘D’ 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 ‘n’ days is given as 1D×n. This is also termed as the Unitary Method.

  • If ‘M’ denotes the number of men, ‘D’ denotes the number of days, ‘H’ denotes the number of hours per day and ‘W’ denotes the amount of work, then the relation among the four can be given as:

(M×D×H)/W=Constant

Where ‘constant’ can be a numeric value of some sort.

  • If ‘M1’ can do ‘W1’ work in ‘D1’ days where he/she works ‘H1’ hours per day and if ‘M2’ can do ‘W2’ work in ‘D2’ days where he/she works ‘H2’ hours per day, then the relation between the two can be given as:

(M1×D1×H1)/W1= (M2×D2×H2)/W2

  • If Ravi is a good worker than Suresh by ‘x’ times, then
  1. Ratio of work done by Ravi and Suresh is given as x:1
  2. 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. 
  • If Ravi and Suresh can complete a given work in ‘p’ and ‘q’ 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. 
  • If Ravi, Suresh and Rajesh can do a piece of work in ‘p’, ‘q’ and ‘r’ days respectively, then together they can finish the work in pqr/(pq+qr+rp) days.
  • If Ravi can do a piece of work in ‘p’ days and Ravi and Suresh together can do the same work in ‘q’ days, then number of days required by Suresh alone to do the same work is pq/(p-q) days. 
  • If Ravi and Suresh can do a piece of work in ‘p’ days, Suresh and Rajesh can do it in ‘q’ days and Rajesh and Ravi can do it in ‘r’ days, then:
  1. Days required when Ravi, Suresh and Rajesh work together is given by 2pqr/(pq+qr+rp).
  2. Ravi alone can do the work in 2pqr/(pq+qr-rp).
  3. Suresh alone can do the work in 2pqr/(-pq+qr+rp).
  4. Rajesh alone can do the work in 2pqr/(pq-qr+rp).
  • Suppose Ravi and Suresh can complete the work in ‘d’ days. If Ravi working alone takes ‘p’ days more than he and Suresh working together and Suresh working alone takes ‘q’ days more than he and Ravi working together, then d= sqrt of pq
  • If a given group of men ‘m1’ and women ‘w1’ can do a given work in ‘D’ days, then another group of men ‘m2’ and women ‘w2’ will take Dm1w1/(m2w1+m1w2) days.
  • If a given group of men ‘m’, women ‘w’ and boy ‘b’ can complete a work in ‘D’ 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 ‘p’, ‘q’ and ‘r’ 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 ‘w1’ amount of work, Suresh did ‘w2’ amount of work and Rajesh did ‘w3’ 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.
  • ‘M’ number of people can do a work in ‘D’ days. If there were ‘m’ people more, then the work can be done in ‘d’ less days. Here, the value of ‘M’ is given as M= m(D-d)/d.
  • ‘M’ number of people can do a work in ‘D’ days. If there were ‘m’ people less, then the work can be done in ‘d’ more days. Here, the value of ‘M’ is given as M= m(D+d)/d.
  • Ravi takes ‘p’ days to do a work and Suresh takes ‘q’ days to do the same work. Both of them started working together, however after ‘a’ number of days, Ravi left. So, the total number of days required to complete the work is given as =q(a+p)/(p+q)