Work refers to the physical or mental effort exerted by an individual to produce goods or provide services in exchange for compensation (wages) or other benefits.
Wage is the monetary payment an employee receives for their labor, typically calculated on an hourly, daily, or piece-rate basis.
Time refers to the duration or period required to complete a specific task or amount of work.
This article will provide you with comprehensive explanations and examples that are easy to follow.
Fundamental Concepts and Formulas
- If a person can do a piece of work in ‘n’ days, then in one day, the person will do ‘1/n’ work. Conversely, if the person does ‘1/n’ work in one day, the person will require ‘n’ days to finish the work.
- Work Equivalence: The work done by people can be written mathematically as the multiplication of the rate at which the work is done and the time taken to complete the work.
If two people work at different rates and for different times, the work done by each can be equated by multiplying their rate of work and the time they worked. So, the total work done in both cases will be the same.
Work done = (Rate of work) x (Time)
Now provided work remains the same,
R1 T1 = R2 T2Now, Rate of Work = (Number of Workers) x (Number of Days)
M1 D1 T1 = M2 D2 T2
where,
M = Number of workers
D = Number of days
R = Rate of Work
- In questions where there is a comparison of work and efficiency, we use the formula below:
M1 D1 H1 E1 / W1 = M2 D2 H2 E2 / W2
where,
H = Number of working hours in a day
E = Efficiency of workers
W = Units of work completed
Total work = No. of Days x Efficiency
- If we increase the number of workers (M) or days (D), the work done also increases proportionally.
- If we increase the number of hours per day (H) or the efficiency of workers (E), it also contributes to completing more work.
For Example: A painter can complete a job in 6 days working alone. If he works with a helper who is twice as efficient, how long will it take them together to complete the whole job?
Solution:
Painter's rate: R1 = 1/6 of the job per day.
Helper's rate: R2 = 2 x R1 = 2 x (1/6) = 1/3 of the job per day (twice as efficient).
Combined Rate: R1 + R2 = 1/6 + 1/3 = 1/2
They complete 1/2 of the job per day.
T = 1/R1+R2 = 2
Time together: 2 days
Conclusion: They will finish the job in 2 days.
Shortcut Tricks for work Problems
This trick can be used:
- To find the efficiency of a person
- To find the time taken by an individual to do a piece of work
- To find the time taken by a group of individuals to complete a piece of work
- Work done by an individual in a certain time duration
- Work done by a group of individuals in a certain time duration
The below image illustrates a shortcut trick for calculating the time taken by two individuals, A and B, to complete a task together, based on their individual work rates.
No. of days:
- A takes 20 days to complete the work.
- B takes 30 days to complete the work.
Efficiency:
- A's efficiency is represented as 3 units of work per day.
- B's efficiency is represented as 2 units of work per day.
Work:
- The total work is calculated as the Least Common Multiple (LCM) of 20 and 30, which is 60 units.
The formula used to find the combined time taken by A and B to complete the work is:
- Time = Total Work / (Efficiency of A + Efficiency of B)
- Substituting the values: Time = 60 / (3 + 2) = 60 / 5 = 12 days.
Trick to calculate combine work rate
The below image demonstrates a shortcut trick for calculating the combined work rate of individuals A, B and C based on their individual work days.
No. of days:
- A + B together take 18 days to complete the work (efficiency 4 units/day).
- A + B together take 24 days to complete the work (efficiency 3 units/day).
- A + B together take 36 days to complete the work (efficiency 2 units/day).
Efficiency:
- The efficiency values (4, 3, 2) represent the work units completed per day by A and B together for the respective time periods.
Work:
- The total work is calculated as the Least Common Multiple (LCM) of 18, 24, and 36, which is 72 units.
- Work done by A: A's efficiency is 1.5 units/day, time taken = 72 / 1.5 = 48 days.
- Work done by B: B's efficiency is 2.5 units/day, time taken = 72 / 2.5 = 28.8 days.
- Work done by C: C's efficiency is 0.5 units/day, time taken = 72 / 0.5 = 144 days.
Work, Wages and Time - Questions and Answers
Question 1 :
To complete a work, a person A takes 10 days and another person B takes 15 days. If they work together, in how much time will they complete the work ?
Solution :
Method 1 :
A's one day work (efficiency) = 1/10
B's one day work (efficiency) = 1/15
Total work done in one day = 1/10 + 1/15 = 1/6
Therefore, working together, they can complete the total work in 6 days.
Method 2 (Short Method):
Let the total work be LCM (10, 15) = 30 units
=> A's efficiency = 30/10 = 3 units / day
=> B's efficiency = 30/15 = 2 units / day
Combined efficiency of A and B = 3+2 = 5 units / day
=> In one day, A and B working together can finish of 5 units of work, out of the given 30 units.
Therefore, time taken to complete total work = 30 / 5 = 6 days
Question 2:
Two friends A and B working together can complete an assignment in 4 days. If A can do the assignment alone in 12 days, in how many days can B alone do the assignment?
Solution :
Let the total work be LCM (4, 12) = 12
=> A's efficiency = 12/12 = 1 unit / day
=> Combined efficiency of A and B = 12/4 = 3 units / day
Therefore, B's efficiency = Combined efficiency of A and B - A's efficiency = 2 units/day
So, time is taken by B to complete the assignment alone = 12/2 = 6 days
Question 3 :
Three people A, B and C are working in a factory. A and B working together can finish a task in 18 days whereas B and C working together can do the same task in 24 days and A and C working together can do it in 36 days. In how many days will A, B and C finish the task working together and working separately?
Solution :
Let the total work be LCM (18, 24, 36) = 72
=> Combined efficiency of A and B = 72/18 = 4 units / day
=> Combined efficiency of B and C = 72/24 = 3 units / day
=> Combined efficiency of A and C = 72/36 = 2 units / day
Summing the efficiencies,
2 x (Combined efficiency of A, B and C) = 9 units / day
=> Combined efficiency of A, B and C = 4.5 units / day
Therefore, the time required to complete the task if A, B, and C work together = 72/4.5 = 16 days
Also, to find the individual times, we need to find individual efficiencies. For that, we subtract the combined efficiency of any two from the combined efficiency of all three.
So, Efficiency of A = Combined efficiency of A, B and C - Combined efficiency of B and C = 4.5 - 3 = 1.5 units / day
Efficiency of B = Combined efficiency of A, B and C - Combined efficiency of A and C = 4.5 - 2 = 2.5 units / day
Efficiency of C = Combined efficiency of A, B and C - Combined efficiency of A and B = 4.5 - 4 = 0.5 units / day
Therefore, the time required by A to complete the task alone = 72/1.5 = 48 days
The time required by B to complete the task alone = 72/2.5 = 28.8 days
Time required by C to complete the task alone = 72/0.5 = 144 days
Question 4:
Two friends A and B are employed to do a piece of work in 18 days. If A is twice as efficient as B, find the time taken by each friend to do the work alone.
Solution :
Let the efficiency of B be 1 unit / day.
=> Efficiency of A = 2 unit / day.
=> Combined efficiency of A and B = 2+1 = 3 units / day
=> Total work = No. of Days x Efficiency = 18 days x 3 units / day = 54 units
Therefore, the time required by A to complete the work alone = 54/2 = 27 days
The time required by B to complete the work alone = 54/1 = 54 days
Question 5:
Two workers A and B are employed to do cleanup work. A can clean the whole area in 800 days. He works for 100 days and leaves the work. B working alone finishes the remaining work in 350 days. If A and B would have worked for the whole time, how much time would it have taken to complete the work?
Solution :
Let the total work be 800 units.
=> A's efficiency = 800/800 = 1 unit / day
=> Work is done by A in 100 days = 100 units
=> Remaining work = 700 units
Now, A leaves and B alone completes the remaining 700 units of work in 350 days.
=> Efficiency of B = 700/350 = 2 units / day
Therefore, combined efficiency of A and B = 3 units/day
So, time is taken to complete the work if both A and B would have worked for the whole time = 800 / 3 = 266.667 days