How to Solve Work Done Problems Using Formula and Solved Examples
Work problems typically occur when two people are painting a house together. You are typically asked how long it takes each individual to paint a house of a comparable size and how long it will take the two of them to paint the house when they collaborate. So, here the concept of work done will be used. In this article, we will learn how we can solve the work done problems.
Work Done Related Important Formula
In time and work, we will learn to calculate and determine the number of hours needed to complete a task as well as the amount of work completed in a specific period of time. We are aware that a person's productivity is closely correlated with the time it takes him to do a task.
For Example: Suppose Shyam can finish a work in 7 days.
Then, work done by Shyam in 1 day $=\dfrac{1}{7}$
Rules:
Suppose if a person A can finish a work in $\mathrm{n}$ days.
Then, work done by $A$ in 1 day $=1 / n^{\text {th }}$ part of the work.
Suppose that the work done by $\mathrm{A}$ in 1 day is $\dfrac{1}{n}$
Then, time taken by $\mathrm{A}$ to finish the whole work $=\mathrm{n}$ days.
Solved Problems on Work Done
Here are some solved problems on work done, through which it can be understood in a better way:
Q1. Piyush and Rahul together can complete a work in 18 days. Piyush alone can do the same work in 24 days. What will be the number of days Rahul alone can complete the whole work?
Ans: Piyush and Rahul can complete the work in 18 days
Piyush alone can complete the work in 24 days
Taking the L.C.M of 18 and 24
L.C.M of 18 and 24 is 72
$\Rightarrow$ One day work of Piyush and Rahul $=\dfrac{72}{18}=4$
$\Rightarrow$ One day work of Piyush $=\dfrac{72}{24}=3$
$\Rightarrow$ One day work of Rahul $=4-3=1$
$\Rightarrow$ Number of days Rahul alone takes to complete the work $=\dfrac{72}{1}=72$
$\therefore$ The number of days Rahul takes to complete the whole work is 72.
Q2. A and $B$ together can do a piece of work in 15 days, while $B$ alone can finish it in 20 days. In how many days can $A$ alone finish the work?
Ans: Time taken by $(A+B)$ to finish the work $=15$ days.
Time taken by B alone to finish the work is 20 days.
$(A+B)$ 's 1 day's work $=\dfrac{1}{15}$
and $B^{\prime}$ s 1 day's work $=\dfrac{1}{20}$
A's 1 day's work $=\left\{(A+B)^{\prime}\right.$ s 1 day's work $\}-\left\{B^{\prime}\right.$ s 1 day's work $\}$
$=(\dfrac{1}{15}-\dfrac{1}{20})=\dfrac{4-3}{60}=\dfrac{1}{60}$
Therefore, A alone can finish the work in 60 days.
Q 3. A can do a piece of work in 25 days and $B$ can finish it in 20 days. They work together for 5 days and then A leaves. In how many days will $B$ finish the remaining work?
Ans: Time taken by $\mathrm{A}$ to finish the work $=25$ days.
A's 1 day's work $=\dfrac{1}{25}$
Time taken by $B$ to finish the work $=20$ days.
B's 1 day's work $=\dfrac{1}{20}$
$(A+B)$ 's 1 day's work $=(\dfrac{1}{25}+\dfrac{1}{20})=\dfrac{9}{100}$
$(A+B)$ 's 5 day's work $(5 \times \dfrac{9}{100})=\dfrac{45}{100}=\dfrac{9}{20}$
Remaining work $(1-\dfrac{9}{20})=\dfrac{11}{20}$
Now, $\dfrac{11}{20}$ work is done by $B$ in 1 day
Therefore, $\dfrac{11}{20}$ work will be done by $B$ in $(\dfrac{11}{20} \times 20)$ days $=11$ days.
Hence, the remaining work is done by $B$ in 11 days.
Practice Questions
Here are practice questions related to work done, through which it can be made in a better way:
Q1. In 300 days, Sanjay finished the school project. If Piyush is 50% more productive than Sanjay, how many days will it take him to finish the identical task?
Ans. 200
Q2. A task can be completed by Sourav and Anshu in 18 days. Anshu and Himanshu can do it in 24 and Sourav and Himanshu can do it in 36 days, respectively. How many days will it take Sourav, Himanshu, and Anshu to complete the task if they collaborate?
Ans. 16 Days
Q3. In 600 days, Sanjay finished the school project. If Piyush is 20% more productive than Sanjay, how many days will it take him to finish the identical task?
Ans. 500
Q4. Piyush, Santosh, and Ramesh are hired as construction workers by a builder on one of his projects. They finish a piece of work in 20, 30, and 60 days, respectively. If Santosh and Ramesh help Piyush every third day, how many days will it take him to do the entire task?
Ans. 15 Days
Q5. A project that Santosh and Prajapati are working on can be finished in 30 days. Santosh put in 16 days of labour, and Prajapati took 44 days to finish it all. How many days would it have taken Prajapati to do the entire project on her own?
Ans. 60 Days
Summary
In this article, we discussed the topic of time and work. Time and work are related concepts. Time is a unit of time, work is an activity done in a given time. Time and work are very important in the field of mathematics. This is because it helps them understand the concept of time and how it can be used to solve mathematical equations. The amount of work you do is related to the amount of time you spend on it. We have understood the topic of time and work perfectly by using some solved problems on work done and time spent.
FAQs on Work Done Problems Explained with Concepts and Methods
1. What are work done problems in maths?
Work done problems are mathematical problems that deal with the rate at which a person or machine completes a task and the time taken to finish it. In these questions, work is usually assumed to be 1 whole unit.
- If a person completes a job in n days, their one day’s work = 1/n.
- Total work = Rate × Time.
- These problems are also called Time and Work problems.
2. What is the basic formula for time and work problems?
The basic formula for time and work problems is Work = Rate × Time. Rearranging gives:
- Rate = Work / Time
- Time = Work / Rate
- One day’s work = 1/5
3. How do you calculate one day’s work?
One day’s work is calculated as 1 divided by the total number of days required to complete the work.
- If a person finishes work in 8 days,
- One day’s work = 1/8.
4. How do you solve work done problems with two people working together?
To solve work problems with two people, add their individual work rates to find the combined rate.
- If A finishes in 6 days → A’s one day work = 1/6
- If B finishes in 3 days → B’s one day work = 1/3
- Together = 1/6 + 1/3 = 1/2
5. What is the formula when men and women have different work efficiencies?
When efficiencies differ, convert everyone’s work into a common work rate before solving.
- If 1 man = 2 women (efficiency ratio 1:2),
- Convert all workers into either men or women units.
- Total work = Combined rate × Time.
6. How do you solve work done problems with pipes and cisterns?
In pipes and cisterns problems, treat filling as positive work and emptying as negative work.
- If a pipe fills a tank in 4 hours → rate = 1/4
- If a drain empties in 6 hours → rate = -1/6
- Net rate = 1/4 − 1/6 = 1/12
7. What is the LCM method in work done problems?
The LCM method assumes total work as the LCM of given time values to avoid fractions.
- If A takes 4 days and B takes 6 days,
- LCM of 4 and 6 = 12 units (total work).
- A’s one day work = 12/4 = 3 units
- B’s one day work = 12/6 = 2 units
8. How do you calculate time when one person leaves the work midway?
When a person leaves midway, calculate work done in parts using their individual rates.
- Find combined rate for the initial period.
- Subtract work completed from total work.
- Divide remaining work by the new rate.
9. What are common mistakes in time and work problems?
The most common mistake in work done problems is confusing time with work rate.
- Not converting days into one day’s work.
- Adding time instead of adding rates.
- Ignoring negative sign in pipes and cisterns.
- Forgetting to subtract completed work when someone leaves.
10. Can you give a simple example of a work done problem?
Yes, here is a simple example: If A can complete a task in 10 days, how long will A take to finish half the work? The answer is 5 days.
- A’s one day work = 1/10
- Half work = 1/2
- Time = (1/2) ÷ (1/10) = 5 days
