We have covered the Time and Work Concepts, and now we will learn the important Time and Work formulas in detail. Under each formula, we will understand its application by solving relevant questions and examples.
This will help you understand not only which formula to use, but also how and where to apply it while solving Time and Work questions.
So, let us start.
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Time and Work Formulas
We have already learnt the Unitary Method in Ratio and Proportion. Now, let us apply the same idea to Time and Work.
Example 1:
If a person takes 10 days to complete a task, how much of the task will be completed in 1 day?
Task completed in 1 day = 1/10 of the task
Similarly, if a helper takes 5 days to complete a drainage work, then in 1 day, the helper will complete:
1/5 of the work
If the same helper takes 15 days to complete the drainage work, then in 1 day, the helper will complete:
1/15 of the work
So, we can derive a simple relation between the work completed and the time taken.
If a person takes D days to complete a task, then the work completed in 1 day will be:
Work completed in 1 day = 1/D of the work
Now, what if we have to find how much work will be done in, say, 5 days? From the above formula, we can easily find it.
Work completed in 5 days = 5 × (1/D) of the work
= 5/D of the work
Example 2:
If Sankirtan can complete a house-cleaning task in 10 days, how much of the work will he complete in 5 days?
Work completed in 5 days
= 5 × (1/10)
= 5/10
= 1/2
Answer: 1/2 of the work
What if the Number of Persons Is Increased?
Now, we have learnt how to find the work completed by one person. But what happens if the number of persons working on the same task is increased?
Example 3:
If a person can complete a task in 15 days, then:
Work completed by 1 person in 1 day
= 1/15 of the task
Now, if one more person joins, there are 2 persons working on the same task. Assuming both persons have the same efficiency, they will complete:
Work completed by 2 persons in 1 day
= 2 × 1/15
= 2/15 of the task
Therefore, the number of days required to complete the whole task will be:
Time required = 1 ÷ (2/15)
= 15/2
= 7.5 days
Answer: 7.5 days
What if there are not 2, but 3 people working to complete the same task with the same efficiency?
The task will be completed in:
15/3 = 5 days
Now we can see that increasing the number of people decreases the number of days required to complete the same task, when all persons have the same efficiency.
Therefore, the number of workers and the number of days are inversely proportional.
Number of workers ∝ 1 / Number of days
Or,
Number of workers (w) × Number of days (d) = Constant
w₁ × d₁ = w₂ × d₂ = w₃ × d₃ = Constant
This applies when the total work remains the same and all workers have the same efficiency.
Now let us apply this formula in the example below:
Example 4:
If 6 workers can complete a task in 20 days, how many days will 10 workers take to complete the same task, assuming all workers have the same efficiency?
Given:
w₁ = 6, d₁ = 20, w₂ = 10, d₂ = ?
Using,
w₁ × d₁ = w₂ × d₂
6 × 20 = 10 × d₂
d₂ = 120/10 = 12 days
Answer: 12 days
Efficiency
You have noticed that in the above examples, it was clearly stated that the workers had the same efficiency. But what if the workers do not have the same efficiency?
Suppose, A can complete a task in 20 days, while B can complete the same task in 25 days.
Work completed by A in 1 day
= 1/20 of the task
Work completed by B in 1 day
= 1/25 of the task
So, we can see that A completes more work in one day than B.
This means A is more efficient than B, because A can complete the same task in fewer days.
So, we can say that efficiency is inversely proportional to the time taken.
Efficiency ∝ 1/Time
Efficiency Ratio of A and B
Efficiency of A : Efficiency of B
= 1/20 : 1/25
= 25 : 20
= 5 : 4
Example 5:
A railway contractor, A, can complete a particular maintenance work in 20 days, while another contractor, B, can complete the same work in 25 days. Who is more efficient and by what ratio?
Solution:
Efficiency of A : Efficiency of B
= 1/20 : 1/25
= 25 : 20
= 5 : 4
Therefore, contractor A is more efficient than contractor B.
Answer: A is more efficient, and their efficiency ratio is 5 : 4.
Example 6:
A railway contractor has assigned two workers, A and B, to complete a track maintenance work. A can complete the work in 12 days, while B can complete the same work in 6 days. If they work together, how much of the work will they complete in 1 day?
Solution:
Work completed by A in 1 day
= 1/12
Work completed by B in 1 day
= 1/6
Therefore,
Work completed by A and B together in 1 day
= 1/12 + 1/6
= 1/12 + 2/12
= 3/12 = 1/4
Answer: 1/4 of the work will be completed in 1 day.
Time Taken When A and B Work Together
Let’s take the above example 6 and derive a relation between time and work when both A and B work together.
Example 7:
In one day, A will complete:
1/12 of the work
And B will complete in one day:
1/6 of the work
So, if A and B work together, in one day they will complete:
1/12 + 1/6
= 1/12 + 2/12
= 3/12
= 1/4 of the work
So, if A and B work together, they can complete the whole work in:
1 ÷ 1/4 = 4 days
Now, suppose A can complete a work in N days and B can complete the same work in M days. In how many days will they complete the work together?
In 1 day, A will complete:
1/N of the work
And in 1 day, B will complete:
1/M of the work
Therefore, A and B together will complete in 1 day:
1/N + 1/M
= (M + N)/MN of the work
If they complete (M + N)/MN of the work in 1 day, then the time required to complete the whole work is:
Time = 1 ÷ [(M + N)/MN]
= MN/(M + N) days
LCM Method in Time and Work
In the above Example 7, you saw that we calculated the work completed by A and B together using fractions:
A’s 1-day work = 1/12
B’s 1-day work = 1/6
So,
1/12 + 1/6
= 1/12 + 2/12
= 3/12
= 1/4 of the work
Here, we used the LCM of 12 and 6, which is 12, to add the fractions easily.
We will cover the LCM method in much more detail in our LCM & HCF Chapter.
Example 8:
A can complete a work in 3 days, while B can complete the same work in 4 days. In how many days will they complete the work together?
A’s 1-day work = 1/3
B’s 1-day work = 1/4
LCM of 3 and 4 = 12
Therefore,
1/3 + 1/4
= 4/12 + 3/12
= 7/12
So, A and B together complete 7/12 of the work in 1 day.
Therefore,
Time required = 1 ÷ 7/12
= 12/7 days
= 1 5/7 days
Answer: 12/7 days or 1 5/7 days
Practice Question on Time and Work Formula
Practice Question 1:
A can complete a work in 9 days, while B can complete the same work in 18 days. In how many days will they complete the work together?
Answer: 6 days
Practice Question 2:
16 workers can complete a work in 15 days. How many workers are required to complete the same work in 12 days, assuming all workers have the same efficiency?
Answer: 20 workers
For more Time and Work practice questions with answers, visit our Time and Work Practice Questions section.
What We Learnt in Time and Work Formulas
- How to calculate work completed in 1 day.
- How to calculate work completed in a given number of days.
- How the number of workers and number of days are related when workers have the same efficiency.
- How to calculate and compare efficiency.
- How to calculate the work completed when two persons work together.
- How to find the time taken when two persons work together.
- How to use the LCM method to simplify fractional work calculations.
Quick Note: Time and Work Formulas
| Concept | Formula |
| Work completed in 1 day | If a person completes a work in D days, 1-day work = 1/D |
| Work completed in D days | Work completed in d days = d/D |
| Workers and Days | w₁ × d₁ = w₂ × d₂ when workers have the same efficiency |
| Workers and Days – Proportion | Number of workers ∝ 1 / Number of days |
| Efficiency and Time | Efficiency ∝ 1 / Time |
| Efficiency Ratio | If A takes N days and B takes M days, efficiency ratio = M : N |
| A and B – 1-Day Work Together | 1/N + 1/M = (M + N)/MN of the work |
| A and B – Time Together | Time = MN/(M + N) days |
| LCM Method | Use the LCM of the given days as total work units to simplify fractional work calculations |