‘Time and work’ is a topic from which a lot of questions are asked in the competitive examinations.

Generally, questions are related to the men doing work individually and the time required in doing the same work together. Then there will be some questions where a group of people starts the work and after some time, some of them leave the work and you are asked to calculate the time required for the remaining persons to complete the work. But how do you go about solving such scenarios? Refer to this article for it!

You will learn how to solve different types of problems related to chapter time and work. But before starting with the types of problems, we will learn some basics about the topic.

**Concept:**

In time and work problems, we assume that the work done by the persons is same on each day unless it is specified in the question that they are doing the work with different rates on different days.

**Example 1: **Let a person A can do the work in 10 days. It means that each day he will do 1/10th of the work. Let another person B can do the same work in 15 days, so B can do 1/15th of the whole work every day. Now, if we employee both A and B to the work then in how many days the work will be completed?

**Solution:**Well!! In time and work problems you just keep two things in mind. First, that every person is doing his own work and second, for how many days each person is working.

Here if we employ A and B to do the work then in one day, A will do his own 1/10th part of the work and B will do his own 1/15th part of the work. So together they will do of the total work in one day. Now as they are doing 1/6th of the work in one day, so the total time taken by them will be 6 days.

In this example we took the whole work as one unit.

There is another approach to solve these questions where fractions are not involved. In that case we take the whole work that is exactly divisible by the total days taken by A and B individually which is normally the LCM of the days.

So,in the above example, the LCM of 10 and 15 is 30, so let the total work is 30 units.

Now as A completes the work in 10 days, so he is doing 3 units work per day. Similarly, B is doing 2 units of the work daily. Together they will do 5 units of work daily.

So, to complete the whole 30 units, they will take 30/5 = 6 days.

Now let us see another problem.

**Example 2: **Let A and B can do a work in 15 days and 20 days,respectively. Both started the work together. After 6 days, A left the work. In how many days B will complete the remaining work?

**Solution:** Here A and B can do 1/15th and 1/20th work in one day working independently.

So together they will do work in one day.

As they did the work together for six days, so in 6 days they will do of the total work.

Now, the remaining work will be done by B alone. B can do 1 unit work in 20 days, so he will do 3/10th work in

Let us try this question with the other method where the work is not 1 unit but a common multiple of the number of days. Let the total work = LCM (15, 20) = 60 units.

So, A can do 4 units in a day and B can do 3 units in a day. Together they can do 7 units in one day.

As they worked together for 6 days, so in six days they will do 7 × 6 = 42 units.

The remaining work is 60 – 42 = 18 units which will be done by B alone and as he can do 3 units in a day so he will take 18/3 = 6 days to complete it.

**Note:** In these questions you should be careful about what is asked. Whether the question is asking about the number of days to complete the remaining work or to complete the whole work? If in the above question we were asked to calculate the number of days required to do the whole work, then the answer would be 6 + 6 = 12 days (6 days when they worked together and 6 days when B worked alone)

**Example 3: **Let A and B can do a work in 20 days and 30 days, respectively. B started the work. After how many days A should join the work so that the work completes in 15 days?

**Solution:** Let the total work is LCM (20, 30) = 60 units.

So, A can do 3 units per day and B can do 2 units per day.

Now we want to complete the work in 15 days and B has already started the work. It means that B will work for full 15 days (Remember you must keep in mind that every person is doing his work and for how many days he is doing).

So, he will do 15 × 2 = 30 units in 15 days.

The remaining work is 60 – 30 = 30 units which will be done by A.

As A can do 3 units so he will take 30/3 = 10 days. It means that A worked for 10 days.

As the total works is completed in 15 days and A worked for only 10 days, so he should join after 15 – 10 = 5 days.

Hope that the above examples have made the concept clear to you all.

Now sometimes instead of the number of days the efficiency of the persons is given in the questions. Let us see how it works.

If A is twice as efficient asB, then you can take its meaning in two ways.

(i) At the same time, A will do twice the work done by B.

(ii) If the work is constant, then A will take half the time taken by B to complete it.

Let us see an example.

**Example 4: **A is twice efficient than B and together they can do a work in 30 days. In how many days they will do the work individually?

**Solution:** In this question, the work is constant.

As A is twice efficient than B so he will take half time than B.

Let the time taken by A alone is ‘x’ days and by B alone is ‘y’ days.

Together they take 30 days to complete the work.

So, we have

So, A can do the work individually in 45 days and B can do it in 2x = 2 × 45 = 90 days.

This question can also be solved by another method:

Since A is twice efficient than B, so let us say that A can do 2 units work in one day and B can do 1 unit work in 1 day. Together they will do 3 units in a day.

Now as the work is completed in 30 days, so the total work = 30×3 = 90 units.

So, A alone will take 90/2 = 45 days and B alone will take 90/1 = 90 days to complete it.

You can also use the following concepts that can directly help you to solve the question.

**Concept 1:**

If A can do a piece of work in x days and B can do it in y days, then A and B working together will do the same work in days.

**Derivation:**

If A can do 1 work in x days,then,

In 1 day, work done by A will be =

If B can do 1 work in y days, then,

In 1 day, work done by B will be =

Their combined work in 1 day = of total work = of total work

Total time taken to do the work =

**Concept 2:**

If A,B, and C can do a work in x,y, and z days respectively then all of them working together can finish the work in days.

**Example 5:** A can do a piece of work in 6 days and B can do it in 5 days. How long will it take if both work together?

**Solution: **Using the formula, the number of days =

**Example 6:** A can do a piece of work in 6 days, B can do it in 12 days, and C can do it in 5 days.How long will they take to complete the work?

**Solution: **Using the formula, the number of days =

**Concept 3:**

If A and B together can do a piece of work in m days and A alone can do it in ndays,then B alone can do the work in

**Derivation:**

(A+B)’s 1 day work =

A’s 1 day work =

B’s 1 day work= of total work

Time taken to do total work =

**Example 7: **A and B together can do a piece of work in 9 days and A alone can do it in 12 days. In how many days B alone can do it?

**Solution: **Using the formula the required time taken by B =

**EXERCISE: **

**Question 1: **A and B can do a piece of work in 72 days. B and C can do it in 120 days. A and C can do it in 90 days. In how many days all the three together can do the work?

(1) 80 days

(2) 100 days

(3) 60 days

(4) 150 days

### Answer and Explanation

**Solution: (3)**

**Question 2: **A and B can do a piece of work in 10 days, B and C in 15 days and C and A in 20 days. C alone can do the work in:

(1) 60 days

(2) 120 days

(3) 80 days

(4) 30 days

### Answer and Explanation

**Solution: (2)**

According to the question,

Work done by (A+ B) in one day = part of work

Work done by (B+ C) together in one day = part of work

Work done by (C + A) together in one day = part of work

By adding all the equations

Hence, C will finish the whole work in 120 days.

**Question 3:** A can do a piece of work in 4 hours; B and C can do it in 3 hours; A and C can do it in 2 hours. How long will B alone take to do it?

(1) 10 hours

(2) 12 hours

(3) 8 hours

(4) 24 hours

### Answer and Explanation

**Solution: (2)**

A can do 1 work in 4 days.

Hence, B alone can do the work in 12 hours.

**Question 4:** While working 7 hours a day, A alone can complete a piece of work in 6 days and B alone in 8 days. In what time would they complete it together, working 8 hours a day?

(1) 3 days

(2) 4 days

(3) 2.5 days

(4) 3.6 days

### Answer and Explanation

**Solution: (1)**

A alone can complete the work in 42 days working 1 hour daily.

Similarly, B will take 56 days working 1 hour daily.

**Question 5: **Ronald and Elan are working on an Assignment. Ronald takes 6 hours to type 32 pages on a computer, while Elan takes 5 hours to type 40 pages. How much time will they take working together on two different computers to type an assignment of 110 pages?

(1) 7 hrs. 30 min.

(2) 8 hrs.

(3) 8 hrs. 15 min.

(4) 8 hrs. 25 min.

### Answer and Explanation

**Solution: (3)**

Ronald takes 6 hours to type 32 pages.

Roland’s 1 hour’s work pages.

Elan takes 5 hours to type 40 pages.

Elan’s 1 hour’s work = 8 pages.

1 hour’s work of both:

pages

To type pages, they take 1hr.

To type 110 page, they will take hours= 8 hours 15 minutes

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**Question 6: **A and B can do a work in 18 and 24 days, respectively. They worked together for 8 days and then A left, the remaining work was finished by B in:

(1) 5 days

(2) 5 days

(3) 8 days

(4) 10 days

### Answer and Explanation

**Solution: (2)**

A’s one day’s work =

B’s one day’s work

(A + B)’s 1 day work =

(A + B)’s 8 day’s work

As we consider total work as 1

∴ Remaining work =

This will be done by B.

∴ Time taken by B to finish of work =1 day

Time taken by B to finish of work =

**Question 7: **A and B can do a piece of work in 28 and 35 days respectively. They began to work together but A leaves after some time and B completed remaining work in 17 days. After how many days did A leave?

(1) 14 days

(2) 9 days

(3) 8 days

(4) 7 days

### Answer and Explanation

**Solution: (3)**

Let A worked for x days.

A’s 1 day work= of work

B’s 1 day work= of work

B works for (x+17) days so

According to the question (A+B)’s combined work will add up to 1

⇒

⇒ 5x + 4x + 68 = 140

⇒ 9x = 140 – 68 = 72

⇒ x = 8

∴ A worked for 8 days.

**Question 8: **A man and a boy can complete a work together in 24 days. If for the last six days man alone does the work, then it is completed in 26 days. How long the boy will take to complete the work alone? (1) 72 days

(2) 20 days

(3) 24 days

(4) 36 days

### Answer and Explanation

**Solution: (1)**

Suppose a man can complete the work in x days and a boy can complete in y days.

1 day work for a man =

1 day work for a boy =

They can complete it together in 24 days.

In 24 days, man’s work =

In 24 days, boy’s work =

**Combined work **

….. (1)

…… (2)

Solving the two equations together,we get y = 72 days.

**Question 9: **A and B together can complete a work in 8 days. B alone can complete that work in 12 days. B alone worked for four days. After that how long will A alone take to complete the work?

(1) 15 days

(2) 18 days

(3) 16 days

(4) 20 days

### Answer and Explanation

**Solution: (3)**

Apply formula: –

Time taken by A to do 1 work =

Here, m = 8 and n=12

So the required time = =24 days

Work done by B in one day = th of work

Work done by B in 4 days => rd of work

Remaining work

part

A can complete a work in 24 days.

A can complete part of work in = 16 days

**Question 10: **A, B and C can do a piece of work in 10, 12 and 15 days respectively. A leaves 5 days before the completion of the work and B leaves 2 days after A. The whole work lasts for

(1) 6 days

(2) 12 days

(3) 13 days

(4) 7 days

### Answer and Explanation

**Solution: (4)**

Required time to complete the work is x.

And all will add up to do 1 work.

∴

⇒

⇒ 15x – 45 = 60

⇒ 15x = 105

⇒ x = 7 days

### Time and WorkQuestions: Problems on Time and Workyou should solve for competitive examination preparation

Welcome to this exercise on Time and Work problems. In this exercise, we build on the basic concepts for Time and Work. As you prepare for your competitive examinations, you will come across questions on Time and Work. Such questions need optimized tackling and can be solved with ease by using simple tricks and understanding the relationships highlighted in this Time and WorkQuestions article. The Time and WorkQuestions exercise comes into the picture where it gives you a chance to practice the highlighted and important concepts related to Time and Workquestion type.