Time and Work

Ankur Kulhari

The 1st and most important thing here to remember is that:
Work is always considered as a unit, that means all the calculations will be done by assuming work as a unit entity.

i.e.

  • ‘Constructing a bridge’ work is considered as a unit task,
  • Filling/emptying a tank or reservoir is considered as a unit entity.

A man can wash 1 cloth in 2 hrs or in 2 hrs the man can wash = 1 cloth

  • From here we can find out the part of cloth washed by man in 1 hr => 1/2
  • Or 1/2 cloth is washed by man in => 1 hr
  • So, 1 cloth can be washed by a man in => 1/(1/2) = 2 hrs.
  • These two things are very-very important in time and work problems.
Some important points to remember:

Work is additive entity, i.e. work can be added directly. (Remember work can be added for same days only)

Example 1. A wall can be constructed by a man in m day, and a woman can construct the same wall in w days. If both works together, in how any days the wall will be constructed?

Solution:
A man takes m hrs to wash = 1 cloth
=> the Man takes 1 hr to wash = 1/m cloth
Similarly, woman takes w hrs to wash = 1 cloth
Woman takes 1 hr to wash = 1/w cloth
So, in 1 hr total washed clothes = washed by man + washed by woman
= (1/m+1/w) clothes
= (m+w/mw) clothes

  • But we have to find in how many hrs. Both together can wash 1 cloth
  • If, (m+w)/mw clothes can be washed in =1 hr
  • 1 cloth can be washed in = 1/[(m+w)/mw] = mw/(m+w)
  • Total time taken in washing a cloth, when both man and woman works together is mw/(m+w).

It is very important to understand this concept. Try to understand it with a real time example considering you as a part of that task.
Suppose you can write 1 page in 1 hr.
And your younger sister can write 1 page in 2 hrs.
if, both of you together have to write that 1 page, how much time it will take?
The 1st thing we can analyze from data is that, time taken to write 1 page together will be < 1hr.
i.e. you can take 1 hr to write = 1 page  {remember, we have to find the time, so time will be taken on LHS and work will be taken on RHS}
And, sister takes 2 hrs. To write = 1 page

  • Sister takes 1 hr to write = ½ page {to bring both you and sister on same place, 1 hr)

=> in 1 hr. How many pages were written ?

  • = 1+1/2= 3/2

Now, we have to find out the time in which 1 page is written by both of you together,
So, work is to be written on LHS,
3/2 page can be written in = 1hr (together)
1 page can be written in = 1/(3/2) = 2/3hrs. {that is less than 1hr, as analyzed by us in the starting}
Once you understand this concept, rest things are very easy now.
Let us look at some rules to be followed in the problems on Time and Work

  1. If A can do a piece of work in n days , then work done by A in 1 day is 1/ n. ie, if a man can paint a wall in 12 days, he paint 1/12th  part of the wall in 1 day.
  2. Similarly if, A does 1/n work in 1 day,
    A does 1 (whole) work in 1/(1/n) = n days.
    ie, if a man paints 1/10th part of a wall in 1 day, then he can paint the whole wall in 10 days.
  3. Efficiency:
    Think in general, Say Ram is very energetic and his brother Shyam is very lazy. Ram can paint a wall in 1 day where as Shyam takes 3 days to paint the same wall.
    Who is more efficient?
    Definitely Ram.
    How can we relate their efficiency with amount of work and number of days taken to complete the work?
    Ram is more energetic, so he can do more work then Shyam in same time
    Ram takes 1 day to paint 1 wall
    Shyam takes 3 days to paint a wall => in 1 day Shyam paints = 1/3 part of wall
    By comparing work done by both in 1 day:
    Ram: Shyam ::1:1/3
    Ram does 3 times more work then Shyam for within same time duration
    We can say Ram is 3 times more efficient then Shyam.
    Similarly, for work Shyam takes 3 times more time then Ram to complete a work.

    • More is the efficiency more work is done in less time
    • Efficiency ἀ 1/Time
    • Efficiency ἀ work
    • If A is thrice as good a workman as B, then ratio of work done by A and B = 3 : 1. ie, if a man works three times as fast as a woman does , then when the work is  complete, 3 parts of the work has been done by the man and 1 part by the woman.
    • If A is thrice as good a workman as B, then ratio of time taken by A and B = 1 : 3. i e, if the woman takes 15 days to complete the work, then the man takes 5 days to complete the same work.
    • If A does a work in ‘a’ days, B can do the same work in ‘b’ days, then A and B together can complete the same work in ab/(a + b) days.
    1. Work ἀ (no of men)
      If the work increases, the no. of men required to do it, also increases,
      i.e., if a man can paint a wall in 2 days, but I want the work to be completed within one day, so I have to hire 2 man for this work to be completed in 1 day (each of them will paint ½ the wall in a day and hence together they will paint complete wall in a day)
      Note: This rule is valid if the work is to be completed in the same number of days.
    2. Men ἀ 1/days
      Men and days are inversely proportional , i e, if the number of men increases , the number of days required to complete the same work decreases and vice versa.
    3. Work ἀ time
      Work and time (days, hrs, min, months, years etc) are directly proportional , ie, if the work increases , the number of days required also increases , if the work is to be completed with the same number of men and vice versa.
    4. Putting them all together:

      • Efficiency (E) ἀ 1/Time {Days and Hours}
      • Efficiency (E) ἀ work (W)
      • Work(W) ἀ (no of men)(M)
      • Men(M) ἀ 1/Time {Days and Hours}
      • Work(W) ἀ Time {Days and Hours}

      E1 = k/D1H1
      E1 = k. W1=> E1/W1 = k
      W1 = k.M1 => W1/M1 = k
      M1=k. 1/D1.H1 => M1.D1.H1 = k
      W1 = k. D1.H1 => W1/D1.H1 = k
      M1.D1.H1/E1.W1 = M2.D2.H2/E2.W2

    Example 1: Ravi  can  do  a   job  10  days .   Determine his   one  day  job.
    Solution.  Ravi’s   10  days   work  =1
    Ravi’s 1 day work = 1/10
    Example 2:  Tinku and Rinku can do a job in 20 days and 30 days alone respectively. If they work together for the job, in how many days they will finish it?

    Sol: in 1 day Tinku can do 1/20 part of the job,

    In 1 day Rinku can do 1/30 part of the job,

    In 1 day both together can do= 1/20+1/30 part of the job = 5/60

    If, 5/60 part of the job can be completed in 1 day,

    So, 1 job can be done in = 1/(5/60) = 60/5= 12 days.

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