Time and Work is an important topic in quantitative aptitude and reasoning. It involves problems related to the amount of work done, efficiency, and the time taken to complete a task.
Key Concepts
1. Work and Time Relationship
If a person completes a task in T days, then in 1 day, the person will complete \displaystyle\frac{1}{T} of the task.
2. Efficiency
Efficiency is inversely proportional to the time taken. For example, if a person A is twice as efficient as person B, A will take half the time B takes to complete the same task.
3. Work Done
\boxed{\text{Work done} = \text{Efficiency} \times \text{Time}}4. Combined Work
If two or more individuals work together, their combined work is the sum of their individual efficiencies.
For example, if A can do a task in T₁ days and B can do the same task in T₂ days, their combined work is:
Work in 1 day = \displaystyle{\frac{1}{T_1}+\frac{1}{T_2}}
Important Formulas
1. Time Taken to Complete a Task Together
If A and B work together and their efficiencies are given, the total time to complete the task is:
\text{Time}=\frac{\text{Total Work}}{\text{A’s Efficiency + B’s Efficiency}}2. Work Left After Partial Work
If a group works for some time and leaves the task incomplete, the work left can be calculated as:
\text{Work Left}=\text{Total Work − Work Done So Far}3. Work with Alternating Shifts
If two individuals A and B work alternately, calculate the work done in 2 days (1 day A and 1 day B) and divide the task accordingly.
How to solve time and work problems?
There are various methods to tackle time and work problems. You can explore all the options and choose the one that works best for you.
Question: A can do a piece of work in 20 days and B alone can do it in 30 days. In how many days A and B can complete the work?
Traditional Method
Total work is assumed to be 1 in this method
In 1 day A does = \displaystyle \frac{1}{20} of work
In 1 day B does = \displaystyle \frac{1}{30} of work
In 1 day A and B together do
\displaystyle \frac{1}{12} of work is done in 1 day
1 work is done in \displaystyle {\frac{1}{\frac{1}{12}}} = 12 days
LCM Method
Total work will assumed as the LCM of given times.
Total work = LCM of 20 and 30 = 60 units
A’s efficiency = \displaystyle \frac{60}{20} = 3 units/day
B’s efficiency = \displaystyle \frac{60}{30} = 2 units/day
Combined efficiency of A and B = 3 + 2 = 5 units/day
To complete 60 units of work, time taken by A and B = \displaystyle \frac{\text{Total work}}{\text{Work per day}} = \displaystyle \frac{60}{5} = 12 days
Using Efficiency Ratio
Total work = efficiency \times time
Let, A’s 1 day work = A
and B’s 1 day work = B
Now, 20A = 30B = Total wok
Now, 20A = 30B
\Rightarrow \displaystyle\frac{\text{A}}{\text{B}}=\displaystyle\frac{3}{2}
If A’s efficiency is 3, B’s efficiency will be 2
Let, time taken by A and B together is x.
Similar Problem: A can complete a task in 12 days, and B can complete the same task in 8 days. How many days will they take to complete the task together? [Ans: \displaystyle 4\frac{4}{5} days]
Question: A and B together can complete a piece of work in 12 days. A alone can do it in 20 days. In how many days B alone complete the work?
Questions: A and B together can do a piece of work in 7\displaystyle \frac{1}{2} days and B alone can do it in 25 days. In how many days can A alone finish the work?
Question: A, B and C together can do a work in 4 days. A and B alone can do it in 12 days and 10 days. How long will C can do it alone?
Question: A and B can complete a work in 12 days, B and C can complete in 20 days, C and A in 15 days.
(i) In how many days A, B and C together can complete the work?
(ii) In how many days A alone can complete the work?
(iii) In how many days B alone can complete the work?
(iv) In how many days C alone can complete the work?
Question: A can do \displaystyle \frac{1}{2} of a piece of a work in 5 days, B can do \displaystyle \frac{3}{5} of the same work in 9 days and C can do \displaystyle \frac{2}{3} of the work in 8 days. In how many days can three of them together do the work?
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