Calculation of Time Taken to Complete a Work is explained in detail here. If you consider a person who finished 1/x of work in one day, then the time is taken by him to finish that work = x days. For example, if a person takes 20 days to complete a piece of work, then the work is done in 1 day = 1/20. Also, if a person completes 1/20th of the work in one day to complete the work, then he will take 10 days. Check out various problems, how to calculate Time to Complete a Work, and objective questions, etc. in this article to get a grip on the complete concept.

Do Read: Problems on Calculating Time

## Time and Work Formula

Consider the work is done by individual persons or groups of persons. Also, the time is taken by the persons to complete the work. The relation between time and work is inversely proportional to each other. That is,

Time = \(\frac{1}{Rate of work}\).

Time is denoted by ‘T’ and work is denoted by ‘W’.

For example: If a person completed the work in 2 hours, then the rate of work is

Rate of work = \(\frac{1}{Time(T)}\).

Rate of work = \(\frac{1}{2}\).

Rate of work = 0.5.

### Quick Methods to Calculate Time to Finish a Work

By learning the basic formulas of time and work, we can easily calculate the time to complete a piece of work. They are

1. Work Done = Time × Rate of work.

. If A can do the work in X days, then the work is done in one day = \(\frac{1}{X}\).

Example: A can do the work in 8 days, then the work done in one day is equal to \(\frac{1}{8}\)th part of the work.

So, the part of the work done by A in D days = \(\frac{D}{X}\)

If A can do the work in 4 days is equal to = 4/8= \(\frac{1}{2}\).

3. Work is done by one person in one day is called the efficiency of the person. Efficiency is expressed in terms of percentage. So, the relation between time and efficiency is inversely proportional. That is,

Time = Number of Days × Efficiency.

Example: A person can complete the work in 6 days by working 12 hours, then the efficiency of the person is

12 = 6 × Efficiency.

Efficiency = 2%.

4. The number of persons M1 can do W1 work in D1 days by working T1 hours every day. Another group of persons M2 can do work W2 in D2 days by working T2 hours every day. So, the relation between them is equal to

\(\frac{𝑀1 ×T1 ×D1}{W1}\) = \(\frac{M2 ×T2 ×D2}{W2}\).

Example: 8 members can make 10 cardboards in 2 days by working 4 hours every day. 6 members can make 12 cardboards by working 8 hours per day. Then the working days of the second group of members are

\(\frac{8 ×4 ×2}{10}\) = \(\frac{6 ×8 ×D2}{12}\)

\(\frac{4}{5}\) = \(\frac{D2}{2}\)

D2 = \(\frac{8}{5}\) = 1.6 days

5. If work done by A is ‘p’ times better than work done by B. So,

The ratio of work done by A and B is equal to p : 1.

The ratio of time to complete the work by A and B is equal to 1 : p.

Example: The work done by A is 2 times better than work done by B. So, the ratio of work done by A and B is equal to 2 : 1. The ratio of time to complete the work by A and B is equal to 1 : 2.

6. X can do the work in x days and Y can do the work in y days. If two persons X and Y are working together, then the work will complete in \(\frac{xy}{x + y}\)

Example: A can do the piece of work in 10 days and B can do the piece of work in 12 days. If A and B working together, then the work will complete in \(\frac{(10(12)}{10 + 12}\).= \(\frac{120}{22}\) = 5 days.

7. X and Y can do the work in 1 day = \(\frac{x+y}{xy}\)

Example: A and B can do the work in 1 day = \(\frac{1}{5}\)th part of the work.

8. X and Y together can complete the work in y days. If X can do the work in x days, then Y can do the work in = \(\frac{xy}{x – y}\) days.

Example: A and B together can complete the work in 10 days. A alone can complete the work in 20 days, then the B can do the work in \(\frac{(10)(20)}{20 – 10}\) = \(\frac{200}{10}\)= 20 days.

9. If X and Y together can complete a work in p days, Y and Z together complete work in q days, Z and X together can complete a work in r days.

- By working X, Y, and Z together, the work is completed in \(\frac{2pqr}{pq + qr + rp}\) days.
- Work done by X alone = \(\frac{2pqr}{pq + qr – rp}\) days.
- Also, work done by Y alone = \(\frac{2pqr}{-pq + qr + rp}\) days.
- Work done by Z alone = \(\frac{2pqr}{pq – qr + rp}\) days.

Example: A and B together can complete a work in 5 days. B and C together can complete the work in 6 days. C and A together can complete a work in 8 days.

(i) By working all together, total work is completed in = \(\frac{2(5)(6)(8)}{(5)(6) + (6)(8)+ (8)(5)}\)

= \(\frac{480}{30 + 48 + 40}\)

= \(\frac{480}{118}\)

= 4 days.

(ii) Total work is completed by A alone is = \(\frac{2(5)(6)(8)}{(5)(6) + (6)(8) – (8)(5)}\)

= \(\frac{480}{30 + 48 – 40}\)

= \(\frac{480}{78 – 40}\)

= \(\frac{480}{38}\)

= 12 days.

(iii) Total work is completed by B alone is = \(\frac{2(5)(6)(8)}{-(5)(6) + (6)(8) +(8)(5)}\)

= \(\frac{480}{-30 + 48 + 40}\)

= \(\frac{480}{88 – 30}\)

= \(\frac{480}{58}\)

= 8 days.

(iv) Total work is completed by C alone is = \(\frac{2(5)(6)(8)}{(5)(6) – (6)(8) +(8)(5)}\)

= \(\frac{480}{30 – 48 + 40}\)

= \(\frac{480}{70 – 48}\)

= \(\frac{480}{22}\)

= 21 days.

10. X can do work in x days and Y is ‘a’ times efficient than X. if X and Y working together, then the work is completed in \(\frac{x}{1 + a}\) days.

Example: A can do a piece of work in 5 days and B is ‘2’ time-efficient than A. If A and B working together, then the work is completed in \(\frac{5}{1 + 2}\) = \(\frac{5}{3}\)= 1.6 days.

11. X and Y together can complete a piece of work in A days and Y is ‘a’ time-efficient than X. So,

- X alone can complete the work in (a + 1) A days.
- Y alone can complete the work in \(\frac{a + 1}{a}\) A days.

Example: A and B together can complete a piece of work in 8 days and B is 2 times efficient than A. Then,

(i) A alone can complete the work in (2 + 1)8 = 24 days.

(ii) B alone can complete the work in ((2+1)/2)8=(3)(4)=12 days.

### Solved Examples on Calculating Time to Complete a Work

We are providing a variety of problems on Time and work concept. Follow the below problems and get the knowledge on the calculation of time to complete a working concept.

**Example 1.**

John can do the piece of work in 4 days and Robert can do the same work in 8 days. In how many days john and Robert complete the work by working together?

**Solution:**

As per the given information, John can complete the work = x = 4 days.

Robert can complete the work = y = 8 days.

The total work is completed by john and Robert = \(\frac{xy}{x + y}\)

Substitute the values in the above equation. Then we will get \(\frac{xy}{x + y}\)

= \(\frac{(4)(8)}{4 + 8}\)

= \(\frac{32}{12}\)

= 2.6 days.

Therefore, John and Robert will complete the work in 2.6 days by working together.

**Example 2.**

Noyal completed the project in 30 days. James is 50% more efficient than Noyal. Find in how many days James will complete the same project?

**Solution:**

As per the information, Noyal can complete the project = x = 30 days.

James work efficiency than Noyal = a = 50%.

So, James alone can complete the project = \(\frac{x}{1 + a}\) days.

Substitute the x and a values in the above equation. Then we get \(\frac{x}{1 + a}\)

= \(\frac{30}{1 + 50%}\)

= \(\frac{(30)(100)}{100 + 50}\)

= \(\frac{3000}{150}\)

= 20 days.

OR

150 : 100 = 3 : 2 = 30 : x

3x = 60.

x= 20 days.

Therefore, James can do a project in 20 days alone.

**Example 3.**

William and Liar together can complete the work in 10 days. William alone can complete the same work in 12 days. Calculate the work done by liar alone?

**Solution:**

As per the given details, William and Liar together complete the work = y = 10 days.

William can complete the work alone = x = 12 days.

Liar alone can complete the work = \(\frac{xy}{x – y}\) days.

Substitute the x and y values in the above equation. We will get as \(\frac{xy}{x – y}\)

= \(\frac{(12)(10)}{12 – 10}\)

= \(\frac{120}{2}\)

= 60 days.

Therefore, Liar alone can do the work in 60 days.

**Example 4.**

Mia completed the given project in 20 days. Find the part of the work done by Mia in 2 days?

**Solution:**

As per the given information, Mia can complete the given project = a = 20 days.

Mia can do the work in 1 day = 1/a = \(\frac{1}{20}\)th part of the work.

So, Mia can do the work in 2 days = \(\frac{2}{a}\)

= \(\frac{2}{20}\)

= \(\frac{1}{10}\)th part of the work.

Finally, Mia can complete the \(\frac{1}{10}\)part of the work in 2 days.

**Example 5.**

John and Ria can do the work in 8 days and 16 days alone. The work started together and John leaves the work before completing it in 2 days. So, in how many days work will be completed?

**Solution:**

As per the details, John can take the time to complete the work = 8 days.

So, John can complete the work in 1 day = \(\frac{1}{8}\)

= \(\frac{x}{8}\)

Ria can take the time to complete the work = 16 days.

So, Ria can complete the work in 1 day = \(\frac{1}{16}\)

= \(\frac{x}{16}\)

But Ria leaves the work before 2 days = \(\frac{x – 2}{16}\)

By working John and Ria together, it will take to complete the work = \(\frac{x}{8}\) + \(\frac{x – 2}{16}\)= 1.

\(\frac{2x + x – 2}{16}\)= 1.

3x-2 = 16.

3x = 18.

x = 6.

Therefore, The total work is completed in 6 days.

**Example 6.**

5 men can finish the work in 5 days and 7 women can also finish the same work at the same time. If 10 men and 7 women are working together, then find the time to complete the work?

**Solution:**

As per the given details, 5 men work is equal to 7 women work. That is, 5 men = 7 women can complete the work = 5 days.

10 men and 7 women are working together. That is,

10 men + 7 women = 10 + 10 = 20 days.

So, by working 10 men and 7 women together, the work is completed in 20 days.

### Frequently Asked Questions on Time and Work

**1. If A can do a piece of work in x days, then what is the work done by A in one day?**

If A can complete a piece of work in x days, then the A can complete a work in 1 day is \(\frac{1}{x}\)th part of the work.

**2. A and B can alone complete the work in x days and y days. If A and B are working together, then what is the work completion time?**

A can complete the work in x days.

A can do the work in 1 day is \(\frac{1}{x}\)

B can complete the work in y days.

B can do the work in 1 day is \(\frac{1}{y}\)

A and B are working together. That is \(\frac{1}{x}\) + \(\frac{1}{y}\) = \(\frac{x + y}{xy}\)

So, A and B together can complete the work in 1 day is \(\frac{x + y}{xy}\)

A and B together can complete the work in \(\frac{xy}{x + y}\) days.

**3. A and B can complete the work in x days, B and c can complete the work in y days, C and A can complete the work in z days. If A, B, and C are working together, then find the time to complete the work?**

A and B can take the time to complete the work in x days.

B and C can take the time to complete the work in y days.

C and A can take the time to complete the work in z days.

A, B, and C are working together. Then the work completion time is equal to

\(\frac{2xyz}{xy + yz + zx}\)

**4. What is the relation between time and work?**

The relation between time and work is inversely proportional. That is

Time = \(\frac{1}{work}\)

Work = \(\frac{1}{Time}\)

**5. What is the relation between work, efficiency, and time?**

The relation between work, efficiency, and time is Work = Efficiency x Time.

If work is more, then the time taken by persons to complete the work is also more and efficiency is also high.