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Time work and efficiency are concepts of mathematics that are interrelated. To understand the concept of time, work and efficiency, it is important to interpret these terms in a literal meaning. All kinds of work require a certain amount of time to complete. The time required to complete a work is called efficiency. To further understand these concepts, sums relating to them should be solved.
Concept of Work and Time
Time and work are simultaneous functions that go hand in hand to estimate efficiency. Work is the effort to complete a task in a certain amount of time.
Work (W) is equal to the amount of time (T) required to finish a task and the rate (R) at which it was completed.
W = RT
Rate and Time are inversely proportional as work is always considered to be 1 unit. The total work is always 1 unit. Therefore,
R = 1/T
T = 1/R
Concept of Efficiency
Efficiency denotes the amount of work that can be done by a person in one day. This concept is often used to compare two workers. A worker’s quality can be determined through efficiency.
If a person is less efficient, the time taken to complete the work would be more
If the time taken to finish a task is less, the person is more efficient
Important terms to solve time-work and efficiency problems
If a person B can finish a work in X days, B’s one day of work according to the formula is, 1/X
If one day’s work of B is 1/X, then the number of days the work would be completed by B is X.
If person B works 6 times better than person A, the work done by B and A is given by: 6:1. The time taken by both B and A is given by 1:6.
Example:
The efficiency of worker S is twice that of T. If S can finish a task 10 days before T, how many days does T need to finish the task?
Solution: Let S can finish the task in x days, and T can finish the task in 2x days, but S can finish the task 10 days before T then,
2x − x = 10
x = 10 days
S can finish the task in
x days = 10 days
and T can finish the task in
2x days = 20 days
The LCM Method
In this method, the total number of work is converted into a number divisible by the number of days the work is required. For example,
If S does a work in 10 days and T does the same job individually in 12 days, in how many days will the work be completed if they work simultaneously?
According to this method, the amount of work can be assumed to be 60 (divisible by both 12 and 10)
For S, the amount of work done individually would be 60/10 = 6units
For T, the amount of work done individually would be 60/12 = 5units
Therefore simultaneously = 11units.
Now according to the formula, work done= Time x Rate
So to complete 60 units of work, it will take 60/11 days.
Work Equivalence
It is a universal concept of measuring work between two sets of people, usually in the same category. The basic assumption in work equivalence is that all workers have the same efficiency. The questions are usually like if 4 men can complete a task in 10 days, how long would it take for 8 men to finish the same task.
In those scenarios, Work = Number of men X number of time taken.
The relation between number of workers, hours, days and the work done is given by:
Example: if 8 men can build 4 units of a well in 16 days, how many units of a well can be constructed in 20 days by 4 men?
Answer: (8 X 16)/4 = (20 X 4)/x
= 32x = 80
x = 2.5
Division of Wages
The salary or wage for any task must be split among the employees according to their contribution to the task’s completion. In other words, the money obtained by finishing a piece of work must be shared according to the amount of effort made. If the workers have worked for the same number of days, the salary can be shared in an equal ratio.
Example:
A can finish a job in 10 days, and B can finish the same job in 15 days. Both A and B work together, however, A only works for a day, and the remaining work is completed by B. How much will B earn if the combined salary is 1000?
Answer:
Now, it is given in the question that A = 10 days , B = 15 days
LCM of (10 , 15) = 30
So efficiency, A= 3 and B= 2
A works for 1 day, which means that the work done by him will be: 3
Work left to be done = 30 – 3 = 27
The work done by B is 27 units.
Remember:
The ratio of work done determines wage.
Ratio of work = 3:27 = 1:9
So the wage will be divided in the same ratio.
Wage of A: Wage of B = 1:9
So B’s share = (9/10) x 1000
= Rs. 900
Pipes and Cisterns
In this concept, the workers have different efficiencies. The only difference is that the effort is measured in terms of filling or emptying a cistern (tank), and the time is measured by how long it takes for the pipe to fill the tank. An input pipe is one that is connected to a cistern that fills it with water. A pipe is called an outlet pipe if it is connected to a cistern that empties it.
Example:
Two pipes A and B can fill a tank separately in 12 and 16 hours, respectively. Let’s assume the tank is empty and if both of them are opened together, estimate the time taken to fill the tank completely.
Part of a tank filled by pipe A in one hour working alone = 1 / 12
Part of a tank filled by pipe B in one hour working alone = 1 / 16
=> Part of tank filled by pipe A and pipe B in one hour working together = (1 / 12) + (1 / 16) = 7 / 48
Therefore, the time is taken to completely fill the tank if both A and B work together = 48 / 7 hours
Conclusion
From the concepts mentioned and the examples provided, it is clear that time, work, and efficiency are interrelated. A piece of work cannot be done without spending a certain amount of time. Work is the amount of time required at a certain rate, whereas efficiency is the rate at which different groups of people or individuals can do a certain job.
