How to solve more Time and Work problems in simpler steps, type 2

Define work amount in terms of days of work and number of work-agents for elegant solution

How to solve time work problems in simpler steps type2

The problems involving Time and work form an important part of most of the competitive job tests such as SSC CGL, Bank POs etc. These problems are taught during middle schools.

To work out these problems, it is usually assumed that a work-agent (say a man) takes certain number of time units T (usually days or hours) to complete the work. So the work rate of the agent in one time unit (a day or an hour) is expressed as $\displaystyle\frac{1}{T}$th portion of the total amount of work.

The reason why this work rate in terms of work portion per unit time is the most important concept in Time and Work problems is - it makes possible summing up of efforts of more than one type of work agents working together at different work rates over unit time. This is the core concept behind the deductions of Time and Work problem of any type.

In conventional approaches, as in any conventiional approach, this core concept is used in a straightforward manner. 

Example: For example, in case of 4 men and 4 women working independently completing a job in 24 days and 12 days respectively, if we are asked to find the number of days taken to complete the job by 4 men and 4 women working together, by the conventional approach, we derive the per day work rate of 1 man and 1 woman as,

$\displaystyle\frac{1}{4\times{24}}=\frac{1}{96}$ portion of work, and $\displaystyle\frac{1}{4\times{12}}=\frac{1}{48}$ portion of work.

When these two teams work together for 1 day, we would now be able to sum up their efforts in one day as,

$\displaystyle\frac{4}{96} + \displaystyle\frac{4}{48} = \frac{3}{24} = \frac{1}{8}$ portion of work.

We now arrive at the desired result using unitary method. The number of days that the two teams would take to complete the job working together would just be inverse of the per day work portion, that is, 8 number of days.

This approach seems to be a bit complex as it deals with inverses, but this is the usual method followed.

First improvement

We have improved this situation in our last post by defining the $A$ as the per day work rate of work agent A, instead of conventional approach of defining, work agent takes $A$ number of days to complete the work and hence, per day work rate of A becomes $\displaystyle\frac{1}{A}$, an inverse of $A$.

By our new definition, we didn't reduce calculations, but increased the comfort level of deductions by eliminating all inverse of variables.

Today we would introduce a second, more powerful technique in dealing with a large portion of Time and Work problems elegantly and quickly.

Second improvement - the Manday technique

For a large number Time and Work problems, a second problem solving strategy makes the process of solving the problems a breeze. We will explain this second special concept by solving two chosen problems. We will as usual show conventional solutions for comparison of effectiveness of the approaches - the conventional approach and the problem solver's approach.

Q1. A work can be completed by 12 men in 24 days and 12 women in 12 days. In how many days would the 12 men and 12 women working together complete the work?

  1. 5 days
  2. 6 days
  3. 8 days
  4. 16 days

Solution 1:

The new technique that we will introduce here is the concept of expressing amount of work in terms of mandays, or woman days, as applicable.

As 12 men complete the job in 24 days, the work amount is $12\times{24}=288$ mandays.

The concept of 1 manday work is simply the amount of work that 1 man will be able to do in 1 day.

This powerful manday concept not only expresses the per day rate of work for 1 man, it also is used for expressing the total amount of work. It serves two purposes. Thus it gains its effectiveness.

For this group of men the total work amount is 288 man days. If 6 such men work, the number of days they would take to finish the work would simply be $288\div{6}=48$ days.

Coming back to our problem, similarly for women, the amount of work is, 144 womandays. It leads to,

144 womandays = 288 mandays,

Or, 1 womanday = 2 mandays.

Or, 12 womandays = 24 mandays.

Thus if 12 men and 12 women work together for a day, it is equivalent to $(12 + 24)=36$ men working on the job for 1 day.

The total work amount being 288 mandays, the number of days taken to complete the job in this case will be, $288\div{36}=8$ days.

Answer: Option c: 8 days.

Key concepts used:

The use of manday concept as a measure of work amount greatly simplifies Time and Work problems on many occasions. The reason of this power in the concept lies in

Combining the days with men in a single valued variable - the work amount. It is a rich concept then, as it combines effect of two variables, the number of men and how many days they take to complete the job.

For example, if 4 men do a job in 10 days, the work amount is 40 mandays. Now if you ask, in how many days 8 men would complete this job, instantly we can find the answer as total work amount (in terms of mandays) divided by number of men resulting in 5 here.

The concept is not only easy to use, it also is intuitive isn't it? If certain number of mandays is divided by a number of men we will definitely get the result as number of days. There would be little scope for any confusion.

In our problem, 12 women do the same work in 12 days. So in the situation of women working, the same work now  amounts to 144 womandays. Equating the two immediately shows that a woman does the job equivalent to 2 men in a day.

Still simpler solution in 20 secs - all in mind, Solution 2

If you analyze the figures quickly you would immediately see that same number of women do the same amount of work in half the time as men. The basic mechanisms in Time and Work problems are,

Number of days to do a work is inversely proportional to number of workers as well as the rate of work. If workers increase in number, they can finish the work in lesser number of days and if work rate of each worker increases that would also result in decreased number of days to complete the work.

Thus we can conclude from the observation that when one woman does double the work of a man, one men's team and one women's team with same numbers together working is equivalent to three men's team working. It means, the combined team will take one third the number of days that 1 men's team took. We don't even have to divide 288 by 36, dividing 24 days by 3 is enough.

If you have noticed, in this solution we have used Abstraction technique and took each unit as 1 work agent, shedding the details of constituent 12 men or 12 women.


Conventional approach - Solution 3

By the problem definition, we can say,

in 1 day 12 men does $\displaystyle\frac{1}{24}$th portion of work, and similiarly,

in 1 day 12 women does $\displaystyle\frac{1}{12}$th portion of work.

Together the two teams do in one day,

$\displaystyle\frac{1}{24} + \displaystyle\frac{1}{12} = \displaystyle\frac{3}{24} =\displaystyle\frac{1}{8}$th portion of work.

Thus the two teams working together would finish the job in 8 days, as before.

Conventional approach - Solution 4

In a less analytical conventional approach the following solution is also possible.

By the problem definition, we can say,

in 1 day 12 men does $\displaystyle\frac{1}{24}$th portion of work. So,

In 1 day 1 man does $\displaystyle\frac{1}{12\times{24}}=\frac{1}{288}$th portion of work.

Similarly, for women,

In 1 day 1 woman does $\displaystyle\frac{1}{12\times{12}}=\frac{1}{144}$th portion of work.

Thus,

In 1 day 1 man and 1 woman working together complete $\displaystyle\frac{1}{288} + \displaystyle\frac{1}{144} = \displaystyle\frac{3}{288}$th portion of work.

And finally,

12 men and 12 women working together complete in 1 day,

$12\times{\displaystyle\frac{3}{288}} = \displaystyle\frac{1}{8}$th portion of work.

That means the two teams, working together, would take 8 days to complete the work.

Please don't say, I have made up this solution by myself.

We will deal with a very typical problem now, where we go on using the manday concept till we reach the solution.

Q2. In 42 days 40 men complete a work. As it happened, instead of all of them working together to finish the job, they started working together, but at the end of every 10th day 5 men left. In how many days would then the work be completed?

  1. 61 days
  2. 63 days
  3. 65 days
  4. 62 days

Solution:

By problem statement the total work amount is $40\times{42} = 1680$ mandays.

Now we will enumerate what happens in every 10 days. But while doing so we look at the choice values using the values as our Free resource and decide to proceed our enumeration till 60th day. Only after the 60th day enumeration we will review the situation.

1st 10 days: total number of days 10: number of days 10 : number of men working 40: work completed = 400 mandays: total work completed 400 mandays.

2nd 10 days: total number of days 20: number of days 10: number of men working 35: work completed = 350 mandays: total work completed 750 mandays.

3rd 10 days: total number of days 30: number of days 10: number of men working 30: work completed = 300 mandays: total work completed 1050 mandays.

4th 10 days: total number of days 40: number of days 10: number of men working 25: work completed = 250 mandays: total work completed 1300 mandays.

5th 10 days: total number of days 50: number of days 10: number of men working 20: work completed = 200 mandays: total work completed 1500 mandays.

6th 10 days: total number of days 60: number of days 10: number of men working 15: work completed = 150 mandays: total work completed 1650 mandays.

So at the end of 60th day or at the beginning of 61st day, 10 men are left and work left to be done is $1680 - 1650 = 30$ mandays.

Thus in 3 more days the whole work would be completed.

Total number of days is then 63 days.

Answer: Option b: 63 days.

Key concepts used:

We have used the Mandays technique to fix the total amount of work at the very beginning.

As the number of men reduces at the end of every 10th day, we decide to use Enumeration technique as the easiest basic approach at this point.

But we also look at the choice values and using our Free resource use principle we decide to review the number of days spent, number of mandays of work left and number of men left at the end of 60th day to easily arrive at the desired result.

We won't show you the solution by conventional approach here.

It is up to you to try to solve this problem in any other way.

We are sure that in this way you will gain more insight into the inner workings of this important topic of Time and Work problems.


Useful resources to refer to

7 steps for sure success in SSC CGL Tier 1 and Tier 2 competitive tests

How to solve Arithmetic problems on Work-time, Work-wages and Pipes-cisterns

Basic concepts on Arithmetic problems on Speed-time-distance Train-running Boat-rivers

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SSC CGL level Solution Set 48 on Time and work in simpler steps 3

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SSC CGL level Solution Set 44 on Work-time Pipes-cisterns Speed-time-distance

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