The relation between time and work is inversely proportional to each other. If the number of persons is more, then they will take less time to complete the work. If the number of persons is less, then the time taken to complete the work is more. We need to calculate the time required to complete a piece of work by a single person or group of persons or machines. In this article, we are providing the number of problems that are related to the calculation of time taken by the persons to complete a piece of work.

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## Time Required to Finish a Piece a Work Sample Problems

Follow the below problems and learn the basic concepts of time required to complete a piece of work.

**Example 1. **

Andrew and Emma both together can finish the work in 10 days. Emma and Liam can complete the work in 15 days. Liam and Andrew can complete the work in 20 days. Find the time required to complete the work by Andrew, Emma, and Liar together and individually?

**Solution:**

As per the given information, Andrew and Emma both can take time to complete the work = a = 10 days.

So, Andrew and Emma can complete the work in 1 day = \(\frac{1}{a}\) = \(\frac{1}{10}\).

Emma and Liam can take time to complete the work together = b = 15 days.

Emma and Liam can complete the work in 1 day = \(\frac{1}{b}\) = \(\frac{1}{15}\).

Liam and Andrew can take time to complete the work together = c = 20 days.

Liam and Andrew can complete the work in one day = \(\frac{1}{c}\)= \(\frac{1}{20}\).

By working all together, work done in one day = \(\frac{1}{a}\) + \(\frac{1}{b}\) + \(\frac{1}{c}\).

(Andrew + Emma) + (Emma + Liam) + (Liam + Andrew) = \(\frac{1}{10}\) + \(\frac{1}{15}\) + \(\frac{1}{20}\).

2(Andrew + Emma + Liam) = \(\frac{6+4+3}{60}\).

Andrew + Emma + Liam = \(\frac{13}{2(60)}\).

Andrew, Emma and Liam together complete the work in one day = \(\frac{13}{120}\).

Andrew, Emma and Liam can take time to complete the total work = \(\frac{120}{13}\) days.

Andrew alone can complete the work in one day = (Andrew + Emma + Liam)’s 1-day work – (Emma + Liam)’s 1-day work.

Andrew’s 1 day work = \(\frac{13}{120}\) – \(\frac{1}{15}\).

L.C.M of 120 and 15 is 120.

Andrew’s 1 day work = \(\frac{13 – 8}{120}\) = \(\frac{5}{120}\).

Andrew alone can take time to complete the work = \(\frac{120}{5}\) = 24 days.

Emma alone can complete the work in one day = (Andrew + Emma + Liam)’s 1-day work – (Andrew + Liam)’s 1-day work.

Emma’s 1 day work = \(\frac{13}{120}\) –\(\frac{1}{20}\).

L.C.M of 120 and 20 is 120.

Andrew’s 1 day work = \(\frac{13 – 6}{120}\) = \(\frac{7}{120}\).

Andrew alone can take time to complete the work = \(\frac{120}{7}\) = 17 days.

Liam alone can complete the work in one day = (Andrew + Emma + Liam)’s 1-day work – (Emma + Andrew)’s 1-day work.

Liam’s 1 day work = \(\frac{13}{120}\) – \(\frac{1}{10}\).

L.C.M of 120 and 10 is 120.

Liam’s 1 day work = \(\frac{13 – 12}{120}\) = \(\frac{1}{120}\).

Andrew alone can take time to complete the work = \(\frac{120}{1}\) = 120 days.

**Example 2.**

James can do a piece of work in 20 days and John can do a piece of work in 15 days. They work together for 5 days and then James stops the work. Calculate the required time to complete the remaining work by John?

**Solution:**

As per the given information, James can complete the piece of work in 20 days. That is a = 20 days.

John can complete the piece of work in 15 days. That is b = 15 days.

So, James can complete the work in 1 day = \(\frac{1}{a}\) = \(\frac{1}{20}\).

So, John can complete the work in 1 day = \(\frac{1}{b}\) = \(\frac{1}{15}\).

James and John both can complete the work in one day = \(\frac{1}{a}\) + \(\frac{1}{b}\).

= \(\frac{1}{20}\) + \(\frac{1}{15}\).

=\(\frac{3+4}{60}\) = \(\frac{7}{60}\).

James and John working together for 5 days = 5×(\(\frac{7}{60}\)) = \(\frac{7}{12}\).

Remaining work is done by John= 1 – \(\frac{7}{12}\) = \(\frac{12 – 7}{12}\) = \(\frac{5}{12}\).

We know that John can complete the work in 15 days.

So, John can complete the \(\frac{5}{12}\)thof the work = \(\frac{5}{12}\)× 15 = \(\frac{25}{4}\)days = 6.25 days

Therefore, John alone can take time to complete the remaining work in 6 days.

**Example 3.**

A can build the wall in 5 days and B can build the wall in 10 days. Find the time to complete the wall by A and B together?

**Solution:**

As per the given information, A can take time to construct the wall = a = 5 days.

B can take time to construct the wall = b = 10 days.

So, A can construct the wall in one day = \(\frac{1}{a}\) = \(\frac{1}{5}\).

B can construct the wall in one day = \(\frac{1}{b}\) = \(\frac{1}{10}\).

If A and B are working together, then they can construct the wall in one day = \(\frac{1}{a}\) + \(\frac{1}{b}\).

= \(\frac{1}{5}\) + \(\frac{1}{10}\).

=\(\frac{a+b}{ab}\) = \(\frac{2 + 1}{10}\) = \(\frac{3}{10}\).

A and B together can take time to construct the wall = \(\frac{ab}{a + b}\) = \(\frac{10}{3}\).

A and B together can construct the wall in (\(\frac{10}{3}\))days.

**Example 4.**

William can fabricate a divider in 15 days, Warner can complete the same work in 30 days. William and Warner work together and get the payment of 5000. So, calculate the time required to complete the work by together and the individual payment of William and Warner?

**Solution:**

As per the given details, William can fabricate a divider in 15 days. That is a = 15 days.

So, William can complete the work in 1 day = \(\frac{1}{a}\) = \(\frac{1}{15}\).

Warner can complete the same work in 30 days. That is b = 30 days.

So, Warner can complete the work in 1 day = \(\frac{1}{b}\) = \(\frac{1}{30}\).

William and Warner both can complete the work in one day = \(\frac{1}{a}\) + \(\frac{1}{b}\)

= \(\frac{1}{15}\) + \(\frac{1}{30}\).

=\(\frac{a+b}{ab}\) = \(\frac{2+1}{30}\) = \(\frac{3}{30}\) = \(\frac{1}{10}\).

William and Warner can take time to complete the work = \(\frac{ab}{a+b}\) = 10 days.

Therefore, William and Warner can complete the work in 10 days

Ratio of the William and Warner work = \(\frac{1}{a}\) : \(\frac{1}{b}\)=\(\frac{1}{15}\): \(\frac{1}{30}\)= 2 : 1

William and Warner both can get the payment of 5000.

William alone can get the payment for work \(\frac{2}{3}\)× 5000 = 2 × 1667 = 3334.

Warner alone can get the payment for work \(\frac{1}{3}\)× 5000 = 1667.

**Example 5.**

Ria is thrice better than Mia at work. Ria can complete the work in 20 days less than Mia. What is the required time to complete the work by both Ria and Mia?

**Solution:**

Let us consider Mia’s work performance as ‘x’.

So, Ria’s work performance is equal to 3x.

So, If Ria completes the work in 1 day, then Mia can complete the work in 3 days.

Therefore, Ria can complete the work in 20 days, then Mia can complete the work in 60 days.

Mia can complete the work in 1 day = \(\frac{1}{60}\).

Ria’s work performance is 3 times better than Mia’s.

So, Ria can complete the work in 1 day = 3\(\frac{1}{60}\) = \(\frac{1}{20}\).

Mia and Ria both can complete the work in 1 day = \(\frac{1}{60}\) + \(\frac{1}{20}\)

= \(\frac{1 + 3}{60}\) = \(\frac{4}{60}\) = \(\frac{1}{15}\).

Therefore, Mia and Ria can take time to complete the work in 15 days.