Time and Work

Time and Work is an important topic in mathematics. It helps us understand how much work a person, machine, pipe, or group can complete in a given time. This concept is useful in school mathematics as well as in competitive exams.

For example, if one person can complete a task in 10 days, then we can find how much work that person can do in one day. Similarly, if two or more people work together, we can calculate how many days they will take to complete the same work.

The main idea of Time and Work is based on efficiency or rate of work.

Basic Concept of Time and Work

If a person completes a work in a certain number of days, then the work done by that person in one day is called one day’s work.

Example

If A can complete a work in 10 days, then:

A’s 1 day work = $\frac{1}{10}$

This means A completes one-tenth of the work in one day.

Formula: 1 day work = $\frac{1}{\text{Total number of days}}$

Important Terms

1. Work

Work means the total task that has to be completed. In most questions, the total work is taken as 1 whole work.

2. Time

Time means the number of days, hours, or minutes required to complete the work.

3. Efficiency

Efficiency means the amount of work done by a person in one unit of time. If a person completes work faster, their efficiency is higher.

For example, if A completes a work in 5 days and B completes the same work in 10 days, then A is more efficient than B.

Important Formulas of Time and Work

Formula 1

If a person completes a work in x days, then:

1 day work = $\frac{1}{x}$

Formula 2

If A can complete a work in x days and B can complete the same work in y days, then their combined 1 day work is:

A + B 1 day work = $\frac{1}{x}$ + $\frac{1}{y}$

Formula 3

Time taken by A and B together:

Time = $\frac{xy}{x+y}$

Formula 4

If A and B together can complete work in x days, and A alone can complete it in y days, then B alone will complete it in:

B alone = $\frac{xy}{y-x}$

Different Types of Time and Work Problems

Type 1: One Person Working Alone

Example 1

A can complete a work in 12 days. How much work will A complete in one day?

Solution

A completes the work in 12 days.

A’s 1 day work = $\frac{1}{12}$

Therefore, A completes $\frac{1}{12}$ part of the work in one day.

Type 2: Two Persons Working Together

Example 2

A can complete a work in 10 days and B can complete the same work in 15 days. In how many days will they complete the work together?

Solution

A’s 1 day work = $\frac{1}{10}$

B’s 1 day work = $\frac{1}{15}$

Together, their 1 day work = $\frac{1}{10}$ + $\frac{1}{15}$

= $\frac{3}{30}$ + $\frac{2}{30}$

= $\frac{5}{30}$ = $\frac{1}{6}$

So, A and B together complete $\frac{1}{6}$ of the work in one day.

Therefore, they will complete the full work in 6 days.

Type 3: Three Persons Working Together

Example 3

A can do a work in 12 days, B can do it in 15 days, and C can do it in 20 days. How many days will they take to complete the work together?

Solution

A’s 1 day work = $\frac{1}{12}$

B’s 1 day work = $\frac{1}{15}$

C’s 1 day work = $\frac{1}{20}$

Together = $\frac{1}{12}$ + $\frac{1}{15}$ + $\frac{1}{20}$

LCM of 12, 15, and 20 = 60

= $\frac{5}{60}$ + $\frac{4}{60}$ + $\frac{3}{60}$

= $\frac{12}{60}$ = $\frac{1}{5}$

So, they will complete the work together in 5 days.

Type 4: Finding Work Done in Some Days

Example 4

A can complete a work in 20 days. How much work will A complete in 5 days?

Solution

A’s 1 day work = $\frac{1}{20}$

A’s 5 days work = 5 × $\frac{1}{20}$

= $\frac{5}{20}$ = $\frac{1}{4}$

Therefore, A will complete $\frac{1}{4}$ of the work in 5 days.

Type 5: Finding Remaining Work

Example 5

A can complete a work in 12 days. How much work will remain after A works for 4 days?

Solution

A’s 1 day work = $\frac{1}{12}$

A’s 4 days work = 4 × $\frac{1}{12}$ = $\frac{4}{12}$ = $\frac{1}{3}$

Remaining work = 1 − $\frac{1}{3}$ = $\frac{2}{3}$

Therefore, $\frac{2}{3}$ of the work remains.

Type 6: Work and Efficiency

Efficiency is the rate of doing work. If a person takes less time, their efficiency is more.

Example 6

A can complete a work in 8 days and B can complete the same work in 12 days. Find the ratio of their efficiencies.

Solution

A’s efficiency = $\frac{1}{8}$

B’s efficiency = $\frac{1}{12}$

Ratio = $\frac{1}{8}$ : $\frac{1}{12}$

Taking LCM 24:

Ratio = 3 : 2

Therefore, the efficiency ratio of A and B is 3 : 2.

Shortcut Method Using LCM

In many Time and Work questions, using LCM makes the solution easier and faster.

Example 7

A can complete a work in 10 days and B can complete it in 15 days. Find the time taken by both together.

Solution

LCM of 10 and 15 = 30

Let total work = 30 units.

A’s 1 day work = $\frac{30}{10}$ = 3 units

B’s 1 day work = $\frac{30}{15}$ = 2 units

Together, they complete = 3 + 2 = 5 units per day

Time taken = $\frac{30}{5}$ = 6 days

Therefore, A and B together complete the work in 6 days.

Type 7: One Person Leaves the Work

Example 8

A and B can complete a work in 12 days. A alone can complete it in 20 days. In how many days can B alone complete the work?

Solution

A and B together 1 day work = $\frac{1}{12}$

A’s 1 day work = $\frac{1}{20}$

B’s 1 day work = $\frac{1}{12}$ − $\frac{1}{20}$

= $\frac{5}{60}$ − $\frac{3}{60}$ = $\frac{2}{60}$ = $\frac{1}{30}$

So, B can complete the work alone in 30 days.

Type 8: Men, Days, and Work

Some questions involve men, days, and amount of work.

Remember:

• More men means less time.
• More work means more time.
• More efficiency means less time.

Formula: Men × Days = Total Work

If work is same, then M₁ × D₁ = M₂ × D₂

Example 9

6 men can complete a work in 10 days. How many days will 12 men take to complete the same work?

Solution

M₁ = 6, D₁ = 10, M₂ = 12, D₂ = ?

M₁ × D₁ = M₂ × D₂

6 × 10 = 12 × D₂

60 = 12D₂

D₂ = $\frac{60}{12}$ = 5

Therefore, 12 men will complete the work in 5 days.

Type 9: Work with Different Efficiencies

Example 10

A is twice as efficient as B. If B can complete a work in 20 days, how many days will A take?

Solution

A is twice as efficient as B.

So, A will take half the time taken by B.

A’s time = $\frac{20}{2}$ = 10 days

Therefore, A will complete the work in 10 days.

Type 10: Pipes and Cisterns

Pipes and cisterns questions are also based on Time and Work. A filling pipe adds water to a tank, while an emptying pipe removes water from the tank.

Example 11

A pipe can fill a tank in 6 hours and another pipe can empty it in 9 hours. If both pipes are opened together, in how many hours will the tank be filled?

Solution

Filling pipe’s 1 hour work = $\frac{1}{6}$

Emptying pipe’s 1 hour work = − $\frac{1}{9}$

Net work in 1 hour = $\frac{1}{6}$ − $\frac{1}{9}$

= $\frac{3}{18}$ − $\frac{2}{18}$ = $\frac{1}{18}$

So, the tank will be filled in 18 hours.

Common Mistakes in Time and Work

1. Adding the number of days directly instead of adding one day’s work.
2. Forgetting to subtract the work of an emptying pipe.
3. Confusing time with efficiency.
4. Not taking total work as 1 or as LCM.
5. Forgetting to calculate remaining work.

Tips to Solve Time and Work Questions

1. Always find one day’s work first.
2. If two or more people work together, add their one day’s work.
3. If someone destroys or empties the work, subtract their work.
4. Use the LCM method for faster calculation.
5. Remember that efficiency and time are inversely related.
6. More workers complete the work in less time.
7. Less workers take more time to complete the same work.

Conclusion

Time and Work is a very useful topic in mathematics. It is based on the simple idea of work done in one day or one hour. Once we understand how to calculate one day’s work, we can easily solve questions involving two or more people, machines, pipes, and workers.

The most important thing to remember is that time and efficiency are inversely related. If efficiency increases, time decreases. If efficiency decreases, time increases.