HomeNon-Routine MathematicsMan-Hours Word Problems / Time...

## Man-Hours Word Problems / Time Work and Men

A man-hour is the work done by one worker in an hour. For time and work-related problems, we assume that all workers or machines are equally efficient which means, every person/ machine finishes equal amount of work in equal span of time unless specified otherwise.

**Work done or task can be expressed in terms of man-hours.**

- Work done/task = number of hrs h taken by m men to complete the task = m x h

a) 1 man working for 3 hours = 1man x 3hrs = 3man-hours

b) 2 men working for 3 hrs = 2men x 3hrs = 6man-hours

- If we double the number of men , twice the work can be done in same time.

a) 2 men working for 1 hour + 1 man working for 3 hrs= (2×1)+(1×3) man-hours

- Work done is directly proportional to number of men m and number of hours h.

a) When both days and hrs are given then hrs = days x hrs per day

__Example__

How much time will two men take to complete a task ,if six men complete the same

task in 12 hrs ?

**Solution:**

__Using Logic__:

If 6 men take 12 hrs then 1/3rd of the men ie two men , will take 3 times the original

time. Therefore, **time required = 3 x 12 =36 hrs**

__Using Formula__:

Task requires 6 men x 12 hrs = 72 man-hrs

Same task can be done by 2 men in h hrs , 2 men x h hrs = 72 man-hrs

**h hrs = 72man-hrs/2 men = 36 hrs**

Two men will take 36 hrs to complete the task.

__Example:__

Twelve machines work twelve hours a day to complete a task .

- How many machines will be required to finish double the task in the same time?
- How many machines will be required to finish the task in same time if four of the machines broke down after 3
^{rd}hour?

**Solution:**

__Using Logic__ :

Let number of machines be y.

a) 12 machines complete the task in 12 hrs. Double the machines are required to finish double the task in the same time. Therefore, number of machines required to complete double the task = 24 .

b) 4 machines worked for 3 hrs = 12 machine hrs – work of one machine in 12 hrs

To finish the task 4 machines should have worked for 12 hrs = 12 X 4 = 48 machine hrs. Therefore, 12 machines + 3 extra machines = 15 machines.

__Using Formula__:

Let number of machines be Y.

a) Task = 12 machines x 12 hrs = 144 machine hrs

Double task = 2 x 144 hrs = 288 machine hrs

288 machine hrs = Y machines X 12hrs

Number of machines Y = 288 machine hrs/ 12 hrs = 24 machines

b) Task = 12 machines x 12 hrs = 144 machine-hrs

144 = (4 machines x 3 hrs ) + (Y-4) machines x 12 hrs

= 12 machine hrs + 12 Y machine hrs – 12 x 4 machine hrs

144 = 12 y machine hrs – 36 machine hrs

180 machine hrs = 12 Y machine hrs

Y= 180 machine hrs/12 hrs = 15 machines

** **

__Example__**: **

A road can be constructed by twelve employees working eight hours a day for thirty days. To speed completion of the project , a contractor deploys 30 employees to complete the project in sixteen days. How many hours a day must the employees work , in order to finish the construction work?

**Solution:**

__Using logic__:

Number of men in 1^{st} case= 12

Number of men in 2^{nd} case=30

Number of men in 2^{nd} case is 2.5 times more than that in 1^{st} case.

Therefore, number of hrs in 1^{st} case will be = 2.5 x number of hrs in 2^{nd} case

i.e. 30 days x 8 hrs = 2.5 (16days x h)

h = 30×8 / 2.5×16 hrs = 6hrs

__Using Formula__:

Construction is done by 12 men in 30 days working 8 hrs a day

Construction of road is Work W = man x hrs = man x (days x hrs)

= 12 men x 30 days x 8 hrs

= 2880man hrs

Construction of same road by 30 men takes h hrs per day for 16 days

W = 2880 man-hrs = 30 men x 16 days x h hrs

h = 2800 man-hrs / (30 men x 16) = 6 hrs