Time and Work: Often, we will hear the quotes like “Time is Money” and “Work hard, and you will succeed”. Do you think there is any correlation between time and work? Of course, there is an inevitable correlation between the two. Time and work deal with the time taken by an individual or a group of individuals to complete a piece of work and the efficiency of the work done by each of them.

Amazing, isn’t it? To know how time and work are dependent on each other and go hand in hand. In this article, we will provide detailed information on time and work. Continue reading to learn more!

Define Time and Work

Work is defined as something which has an effect or outcome. When it is said that someone has done any work, it means he/she has done \(100\% \) of the work. Hence, if Manu finishes his work in \(5\) days, it means that in \(5\) days, he will finish \(100\% \) of the work. Hence, we can say that in \(1\) day he finishes \(20\% \) of the work. Similarly, in \(2\) days, he finishes \(40\% \) of the work, and in \(3\) days, he finishes \(60\% \) of the work.

In our daily lives, we come across two types of variations, direct variation and indirect variation. These are also called direct proportion and inverse proportion.

When two quantities \(x\) and \(y\) are in direct proportion, both \(x\) and \(y\) increases or decreases at the same rate. When \(x\) increases, \(y\) also increases, keeping the ratio between them constant always.

\(x \propto y\)

Removing the proportionality, we can write this as:

\(x = ky,\) where \(k\) is the proportionality constant. \( \Rightarrow \frac{x}{y} = k\)

However, in our daily lives, we come across situations where an increase in one quantity results in a corresponding decrease in another corresponding amount or vice versa. For example, when a motorist travels at a higher speed, the time taken to cover a certain distance is reduced. This is the indirect proportion or indirect variation.

If we consider two quantities \(x\) and \(y,\) then when \(x\) increases, \(y\;\) decreases.

\(x \propto \frac{1}{y}\)

Removing the proportionality, we can write this as \(xy = k,\) where \(k\) is the proportionality constant.

Facts of Time and Work

Time and work relate to indirect proportion. An increase of one quantity leads to the decrease of the other quantity and vice versa.

The more hours people work, the less time it takes to complete the same amount of work. Conversely, the fewer men to work on a particular task, the more time will be taken to complete the job.

Let us understand with the help of an example.

If \(30\) workers can build a wall in \(24\) hours, how many workers will be required to do the same work in \(18\) hours?

Solution: Let the number of workers be \(x.\)

Number of Hours	\(24\)	\(18\)
Number of Workers	\(30\)	\(x\)

We know that the more the number of workers, the lesser time will be taken to complete the same amount of work.

Therefore, the number of hours and the number of workers are in inverse proportion.

So, \(30 \times 24\; = \;x\; \times 18\)

Thus, \(x = \;\frac{{30 \times 24}}{{18}}\; = \;40\) Hence, to finish the work in \(18\) hours, \(40\) workers will be required.

Time and Work Tricks

1. If \(W\) can complete \(\frac{1}{n}\) of work in \(1\) day, then \(W\) will take \(n\) days to complete the whole work.

2. If \(W\) can do a work in days, then \(1\)-day work of \(W = \;\frac{1}{n}.\)

3. If \(W\) and \(Z\) can complete a piece of work in \(x\) and \(y\) days respectively, then if they both start working together, they can finish the same work in \(\frac{{x \times y}}{{x + y}}\) days.

4. If \(W\) completes \(\frac{1}{{{n^{t{\rm{h}}}}}}\) of work in \(x\) hours, then to finish the complete work \(W\) will take \(n\; \times \;x\) hours.

Once you are thorough with the simple tricks and their application, the problems related to time and work will become extremely easy to solve. And, thus, we can say that,
1. The number of men and the amount of time required to complete one work are inversely proportional to each other.
2. Time and work are directly proportional to each other. Thus, in more time, more work can be completed and vice versa.

3. The work can be divided into equal parts, i.e. if a person can complete a piece of work in \(5\) days, then one day work of that person will be \(\frac{1}{{{5^{{\rm{th}}}}}}\;\) part of that work.

Solved Examples on Time and Work

Q.1. Priya can do a piece of work in \(12\) days, and Joyita can do the same work in \(18\) days. How many days will they take to do the work together?
Ans: Priya can do a piece of work in \(12\) days.

Therefore,Priya’s \(1\)-day work \( = \;\frac{1}{{12}}\) of the work

Joyita can do a piece of work in \(18\) days.

Therefore,Joyita’s \(1\)-day work \( = \;\frac{1}{{18}}\) of the work

Thus, Priya’s and Joyita’s \(1\)-day work \( = \;\frac{1}{{12}}\; + \;\frac{1}{{18}}\)

Taking the \({\rm{LCM}}\) of denominators, we get

\(\frac{1}{{12}}\; + \;\frac{1}{{18}} = \frac{{3 + 2}}{{36}}\)

\( = \;\frac{5}{{36}}\) of the work Hence, to finish the piece of work together, they need \(\frac{1}{{\frac{5}{{36}}}}\; = \;\frac{{36}}{5}\) days i.e., \(7\frac{1}{5}\) days.

Q.2. \(6\) workers can pack \(1500\) boxes in \(4\) days. Working at the same rate, how many boxes can \(8\) workers pack in \(6\) days?
Ans: \(6\) worker takes \(4 – \)days to pack \(1500\) boxes.

\(1\) worker takes \(4\)-days to pack \(\frac{{1500}}{6}\) boxes.

\(8\) workers take \(4\)-days to pack \(\frac{{1500}}{6} \times 8 = 2000\) boxes.

\(8\) workers take \(6 – \)days to pack \(\frac{{2000}}{4} \times 6 = 3000\) boxes. Hence, \(8\) workers can pack \(3000\) boxes in \(6\) days.

Q.3. It takes \(42\) days for Manu to construct a \({\rm{12\;m}}\) long wall. How long will it take him to complete a \({\rm{20\;m\;}}\) long wall?
Ans: Time required to construct \({\rm{12\;m\;}}\) long wall \( = \;42\) days

Time required to construct \({\rm{1\;m}}\) long wall \( = \;\frac{{42}}{{12}}\) days

Therefore, time required to construct \({\rm{20\;m}}\) long wall \( = \;\frac{{42}}{{12}}\; \times \;20\; = \;70\) days. Hence, it will take \(70\) days for Manu to complete \({\rm{20\;m}}\) long wall.

Q.4. If \(15\) workers can build a wall in \(48\) hours, how many workers will be required to do the same work in \(20\) hours?
Ans: Let the number of workers be \(x.\)

Number of Hours	\(48\)	\(20\)
Number of Workers	\(15\)	\(x\)

We know that the more the number of workers, the lesser time will be taken to complete the same amount of work.

Therefore, the number of hours and the number of workers are in inverse proportion.

So, \(15 \times 48\; = \;x\; \times 20\)

Thus, \(x = \;\frac{{15 \times 48}}{{20}}\; = \;36\) Hence, to finish the work in \(20\) hours, \(36\) workers will be required.

Q.5. Rachana can do work in \(15\) days and Swati in \(20\) days. If they work on it together for \(4\) days, then what fraction of the work is still left?
Ans: Rachana can do work in \(15\) days.

Therefore, Rachana’s one day work \( = \frac{1}{{15}}\)

Swati can do the same work in \(20\) days.

Therefore, Swati’s one day work \( = \frac{1}{{20}}\)

Swati’s and Rachana’s \(1\)-day work \( = \left( {\frac{1}{{15}} + \frac{1}{{20}}} \right) = \frac{7}{{60}}\)

Swati’s and Rachana’s \(4\)-days work \( = \frac{7}{{60}} \times 4 = \frac{7}{{15}}\)

Therefore, remaining work \( = 1 – \frac{7}{{15}} = \frac{{15 – 7}}{{15}} = \frac{8}{{15}}.\) Hence, \(\frac{8}{{15}}\) part of work is still left.

Summary

Work is described as something which has a result or effect or outcome. Time is defined as when a particular event happens or has occurred or will occur. Time and work are directly proportionate to each other. Furthermore, one can state that in more time, more work can be completed and vice versa. However, it is also important to note that the number of men and the amount of time required to complete one work are inversely proportional to each other

FAQs on Time and Work

Q.1. Define work with the help of an example.
Ans: Work is defined as something which has an effect or outcome. When it is said that someone has done work, it means he has done \(100\% \) of the work. For example, if a person finishes work in \(10\) days, it means, in \(10\) days, he will finish \(100\% \) of the work. Hence, we can say that in \(1\) day, he finishes \(10\% \) of the work or \(\frac{1}{{{{10}^{{\rm{th}}}}}}\) of the work.

Q.2. Are time and work directly proportional to each other?
Ans: Yes, time and work are directly proportional to each other. Thus, in more time, more work can be completed and vice versa.

Q.3. What is the relation between time and work?
Ans: If a piece of work is done in \(n\) number of days, then the work done in one day is \( = \;\frac{1}{n}.\) Work done \( = \) time taken \( \times \) rate of work.

Q.4. If I need to study \(4\) chapters in Mathematics, how many chapters can I study in \(1\) hour?
Ans: I need \(10\) hours to study \(4\) chapters. So, in \(1\) hour, I can cover \(\frac{4}{{10}}\; = \;\frac{2}{5}\) of the chapters.

Q.5. If \(100\% \) of the work is completed in \(1\) day, how much work is done is \(1\) hour?
Ans: We know that \(1\) day \( = \;24\) hours.
Now, in \(24\) hours, a person can complete \(100\% \) of work. In \(1\) hour, a person can complete \(\frac{{100}}{{24}}\; = \;\frac{{25}}{6}\% \) of the work.

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