*Wednesday, June 12th, 2019*

Months have passed, Days are passing by swiftly and with each day you are progressing closer towards CAT exam. As hours, minutes and seconds pass we cannot afford to lose a single second recklessly. In short, Time is running out and we still need to do a lot of work! Time and work is something that always runs in parallel with each other. We are always in short of time as compared to the work to be done. Today I will tell you how to manage your little time and paramount of work problem and this also will serve the relevant purpose of writing this blog i.e. Time and Work problems in quantitative aptitude section of CAT. Yes seriously, Time and Work problems of CAT can teach you how to manage your time for preparing for CAT as well. Think about it, and that’s its purpose of being a part of CAT syllabus. As future business managers, we all need to understand the significant correlation between time and work and must deal with deadlines. We need to understand and figure out how much time is required to finish the given task and with the given level of efficiency of resources etc. That’s why this topic holds a lot of importance as this just not a confusing concept in quantitative aptitude section of CAT, it could be applied to other sections too. As CAT is all about problem-solving in the time constraint. This will fulfill dual purpose one is that we will learn how to solve them in the exam and the second is, we will know how to manage our own time. I know for many of you this topic can become very baffling and difficult to understand. But honestly, this is not as complicated and puzzling as it seems to be. Let’s make this concept easy to follow and grasp firstly as a part of Quantitative Aptitude and sideways I will tell you how can we apply in our other sections

Now there can be hundreds of different combinations in work time problems. I will try to develop this concept step by step by taking one example and adding & changing its scenarios. You can yourselves then make many other such scenarios and practice these problems. So, let’s begin.

Consider, Tom, a young carpenter who owns a furniture shop in a local market and has been given an order to make 4 chairs in 5 days. Now, how much work is he should complete in 1 day to produce the order in time? Or we can also be asked how many chairs he’s able to make in a single day?

The answer to this question is simple using the unitary method if a work is completely done by Tom in 5 days then, the amount of work he does in one day is 1/5^{th} of total work. Visualize it like this, if we have a pizza (as the amount of work to be done) cut into 5 equal parts (no. of days required to complete the given work) then one slice of pizza would represent work done in a single day and that would be 1/5.

Hence, if we are given total no. of days required to complete a task say **‘n’** days then we can easily calculate the amount of work done in a single day as **‘1/n’**. Now if we want to know how many chairs are made in a single day we simply multiply 4*(1/n) and we get 0.8 i.e. he completes 80% of a chair in a single day.

We can also look the above problem as if Tom is able to complete 1/5^{th} of the order he has been given then he is able to complete the entire task in 1/ (1/5) i.e. 5 days.

Alternatively consider, if you are given 10 questions of similar type of say seating arrangement in Logical Reasoning section of CAT and you are given 15 minutes to solve these questions, therefore, no. of questions you can solve in a single minute is 10/15 i.e.66.67% of a question and if you reciprocal this figure you get no. of minutes to solve a single question that is, 1.5 minutes/ question and in this way if you get to know the average time you need to solve a particular kind of problem so that you can effectively divide your time in the exam.

Now let’s add more complexities to above problem such as if tom hires another carpenter named Jon to help him to finish his orders on time as he’s getting multiple orders and he’s is unable to deliver them on time. Suppose Tom receives an order to build 10 chairs in 7 days and if he independently works alone and finishes the order in 12 days and Jon completes the same task in 10 days if he works alone. Will they be able to deliver the order on time if they combinedly do the job?

Let’s find out.

To find out the answer to this question we first need to know how many days both will take if they work together but independently. And to get the answer we calculate work done by them in a single day.

Since Tom single-handedly completes the job in 12 days so work was done by him 1 day = 1/12.

Similarly, Jon completes the entire order in 10 days so work done by him in 1 day = 1/10.

Now if they work jointly but each working on their own in a 1 day would be = 1/12 + 1/10 (work was done by Tom independently + work done by Jon independently)

= 11/60 per day.

No. of days they will take jointly to finish the order would be = 1/ (11/60) = 60/11= 5.45 days. Thus, they will able to deliver the order on time. Similarly, if there are multiple workers such as 3, 4 or more we can follow a similar procedure to find out how much time will they take to complete the job if they work together. There can be many variations such as if Tom, Jon, and Smith all three combinedly performing a task and completes it 5 days and Jon single-handedly does it 12 days, same way Smith finishes the job alone in 15 days. How many days will Tom take to complete the task alone?

To ascertain the answer simply proceed this way

Thus no. of days Tom will complete the task solely would be 1/ (1/20) = 20 days.

Examine another scenario such as if tom starts to bring his 8-year-old son Bran to work and he’s a notorious kid who destroys whatever is been made by Tom and Jon. If Tom gets an order to make 3 wardrobes and Tom can make them solely in 20 days, Jon finishes them alone in 24 days and Bran destroys the work done by both in 15 days. Then how much time will it take by Tom and Jon to complete the order?

This question can also be done similarly on the previous question

I hope you now you can find out the answer to this question on your own by simply plugging all the values in above equation. What if there’s another constraint in the above problem i.e. if they both work 6 hours a day each in an analogous situation then how long will they both take to finish the order given Bran continues to destroy their work for all 6 hours.

To ascertain the solution just find out no. of hours work by each of them in a day i.e.

Tom can complete work in 20*6 = 120 hours. => Work done by Tom in 1 hour = 1/120

Jon can complete work in 24*6 = 144 hours. => Work done by Jon in 1 hour = 1/144

Bran can destroy work in 15*6 = 90 hours. => Work destroyed by Bran in 1 hour = 1/90

Both will finish the work in 240 hours. To find out no. of days when they work 6 hours a day each = 240*1/6 = 40 days.

Both will finish the work in 240 hours. To find out no. of days when they work 6 hours a day each = 240*1/6 = 40 days.

Let’s modify the situation again and suppose now instead of no. of days each Carpenter works solely, efficiency is given such that jon is twice as efficient as Tom in making tables and Tom is thrice as efficient as Jon in making chairs. Tom receives an order to construct 2 tables and 10 chairs. They jointly complete tables in 6 days and chairs in 9 days. Find out how many days will Tom take to individually complete both chair and tables.

Now in the above question, there are

- Two products: Chairs and Tables
- Each workman has different efficiency in relation to the product.
- To deduce the no. of days taken by Tom to complete the task we could solve the questions step by step as given below.

Firstly, assume the efficiency of Tom = x and,

the efficiency of Jon = y.

In case of Tables,

Both can complete the work in 6 days. Hence work done by them in a day is

x + y = 1/6

Also, given Jon is twice as efficient as Tom in making tables. Therefore,

y = 2x

Solving both the equations simultaneously, we get

x = 1/18, y = 1/9

Thus, Tom takes 18 days to make tables solitarily whereas Jon takes 9 days solitarily.

In case of Chairs,

Both can complete the work in 9 days. Hence work done by them in a day is

x + y = 1/9

Also, given Tom is thrice as efficient as Jon in making tables. Therefore,

x = 3y

Solving both the equations simultaneously, we get

x = 1/12, y = 1/36

Thus, Tom takes 12 days to make chairs solitarily and Jon does it in 36 days on its own.

Hence, Total no. of days taken by Tom to wind-up the entire order solely is 18 + 12 = 30 days.

Now consider another scenario where tom receives an order to produce 20 chairs. He promised to complete the job in 25 days solely. He works for 10 days and falls sick. To fulfill the job, he requests his brother who lives uptown to do his work. He agreed and finished the remaining work in 21 days single-handedly. How many will more days be taken by Tom’s brother to complete the entire job solely?

To solve such questions, we need to segregate the question in parts and then solve them individually i.e.

- We will first begin with finding out the amount of work done by Tom in 10 days
- Then, we will calculate the amount of remaining work to be done.
- Afterwards, we will determine no. of days taken by his brother to fulfill the leftover work.

So, let’s do it.

Since Tom can complete the job in 20 days. Work was done by him in a day = 1/25

He works for 10 days. So, work done by him in 10 days = 10*1/25 = ⅖

The work left to be done = 1- ⅖ = ⅗

Now Tom’s brother completes ⅗ of work in 21 days. Therefore, no. of days he will take to complete the entire task = 21* 5/3 = 35 days.

No. of extra days taken by Tom’s brother to complete the job in comparison with Tom = 35-25 = 10 days.

There could be numerous other scenarios and different combinations of above kind of problems. I have taken two individuals Tom and Jon, there can be 3 or more. Efficiency can also be given in percentage instead of absolute terms etc. Also, instead of a man and work these kinds of questions can be asked in form of pipes and cisterns or water tanks and tap, etc.

I will give you an example of one such type of question. You can proceed in an equivalent way as in above questions. It’s just a bit language would be changed.

Consider, a water tank has 10 pipes. Some of them fill the tank other empties it. If each of the pipes can solely fill the tank in 10 hours and each other pipes can alone empty the tank in 5 hours. If the tank is full and all pipes are opened it takes 5 hours to empty the tank fully. Then how many pipes are empties the tank?

Again, to answer this problem we will first assume,

let x be no. of pipes that fill up the tank.

And y be no. of them that empty the tank.

x + y = 10

Therefore, each pipe will fill 1/10^{th} tank per hour => x pipes will fill x*1/10^{th} of tank/hour.

Similarly, each pipe empties tank in 1/5^{th} tank per hour => y pipes will empty y*1/5^{th} of tank/hour.

x/10 – y/5 = -1/5

Solving both the equations simultaneously, we get

x = 6, y = 4.

Thus, there are 4 pipes that empty the tank.

I hope you have understood now how to solve problems on Time and Work. See they are so simple and easy. Just follow the right steps and you will smoothly solve the question. The trick is to find work done in per unit of time. Form equations, plug in the values, solve the equations and then what? Simple, you get the solution. Problem solved!

As I told you and even gave you an example you can use this method to plan your schedule to prepare for CAT and effortlessly solve Time-management problem. This technique can really help you a lot in the exam too to save and manage time because “*Time and Tide wait for none”.*

*All the best!*

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