# Time-Work Equivalence

The essence of time-work equivalence lies in the fact that it exhibits the most fundamental relationship between the three factors as mentioned above viz., work, time and the agent which is completing the work. That is,

Work done = Number of days Ã— Number of men

W = M Ã— D

This gives us an important concept of man-days.

Suppose there are 20 persons working for 10 days to complete a job, then the total work done is equal to 200 man-days. Now, if we change the number of days in which the work is to be completed, then the other factor, i.e., the number of persons will change accordingly, so that, the product of the factors becomes equal to 200 man-days. Product-stability-ratio (Chapter 3) is a very effective tool to calculate this.

Suppose there are 20 persons working for 10 days to complete a job, then the total work done is equal to 200 man-days. Now, if we change the number of days in which the work is to be completed, then the other factor, i.e., the number of persons will change accordingly, so that, the product of the factors becomes equal to 200 man-days. Product-stability-ratio (Chapter 3) is a very effective tool to calculate this.

Example

Seven persons can clean 7 floors by 7 mops in 7 days. In how many days can 5 persons clean 5 floors by 5 mops?

Solution

This problem can be solved through several methods.

**Method 1**To clean 7 floors, we need to have 7 Ã— 7 = 49 man-days.

So, to clean 5 floors, we need to have 35 man-days.

So, 35 = D Ã— 5. So, D = 7 days

**Method 2**Using ratio proportion, less work and less men are involved here.

So, the number of days =

**Method 3**Let us try to have a mental image of this situation: There is a building with seven floors namely F

_{1}, F

_{2},â€¦, F

_{7}and seven persons P

_{1}, P

_{2},..., P

_{7}are cleaning this building in such a way that one floor is being cleaned by each one of them. Since it takes 7 days to complete the whole work, it can be inferred that everybody is taking 7 days to clean his respective floor. So, if there are just floors and five persons are cleaning these five floors, then it will take them seven days (assuming that the top two floors have been demolished).

Now, depending upon different situations, three conditions are possible, in the relationship W = MD (where W = quantity of work, M = number of persons, D = number of days)

# Condition 1: W is constant

M Ã— D = Constant

M Î± 1/D

It can be observed that if the work done is constant, then the number of persons is inversely proportional to the number of days, which means that the multipliers of M and D will be reciprocal. Extending this situation, if 10 persons can do a work in 20 days, then 5 persons can do the same work in 40 days, or, 20 persons will do the same work in 10 days. Further it can be summarized as

w | = | M | Ã— | D |

200 | = | 10 | Ã— | 20 |

200 | = | 5(10 Ã— 1/2) | Ã— | 40(20 Ã— 2) |

200 | = | 20(10 Ã— 2) | Ã— | 0(20 Ã— 1/2) |

Thus, it can be said that the multiplier of M and multiplier of D are reciprocal to each other.

It can be seen with the help of the graph given below
50% 50 40...(i)
Rest 50% (50 + M) 10 ...(ii)
Since work is constant in both the cases, so, the number of men and the number of days will be reciprocal to each other. As the number of days left in (ii) is 1/4th of initial period (i), so the number of persons will become 4 times of the initial number of persons.
Hence, the number of persons = 50 Ã— 4 = 200. So, M = 150 men.

Example-1

Yadavjee contractor undertakes to get a work done in 50 days by 50 labourers. After 40 days, he realizes that only 50% of the work is done. How many more men should be employed so that the work is complete on time?

Solution

W = M Ã— D

Example-2

In the above question, if the schedule can go behind by 10 days, then how many extra men are required to complete the work?

Solution

So, now we will have to complete the work in 60 days.
W = M Ã— D
50% 50 40.....(i)
Rest 50% (50 + M) 20 .....(ii)
Since work is constant in both the cases, so the number of men and the number of days will be reciprocal to each other. As, the number of days left in (ii) is 1/2 of initial period (i), so the number of persons will become 2 times of the initial number of persons.
Hence, the number of persons = 50 Ã— 2 = 100. So, M = 50 men.

Example-3

In question 2, he realizes after 40 days that work is only 20% complete. How many extra men should be employed now so that the work is completed on time?

Solution

W = M Ã— D
20% 50 40 .....(i)
20% 200 10 .....(ii)
So, 80% 800 10 .....(iii)
Hence, 750 more persons are needed to complete the job on time.

# Condition 2: D is constant

W Î± MMore work will be done if we employ more men and vice versa. It means that multiplier of W and M will be same. It can be seen with the help of the graph given below

# Condition 3: M is constant

W Î± DIn general, we can summarize that

Example-4

12 persons can cut 10 trees in 16 days. In how many days can 8 persons cut 12 trees?

Solution

Here W

_{1}= 10 W_{2}= 12M

_{1}= 12 M_{2}= 8D

_{1}= 16 D_{2}= ?Putting the values in the equation W

_{1}/W_{2}= (M_{1}/M_{2}) Ã— (D_{1}/D_{2})We get