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


First Time and Work rich concept we have used was, 

Define Worker variables in terms of Work rate, that is, work done by the worker in a Day (assuming Day as the time unit).

This definition enabled us to directly add the work done by two workers when they work together, transferring the fractions onto the Work W.

Second Time and work rich concept of Mandays

We define work amount as 480 mandays for example, if 48 men complete the work in 10 days.

First advantage of using this concept is, we can deduce any combination of number of men and number of days, applying factorization concepts on the total work mandays.

In this case of 480 mandays, we can easily say, the work will be completed by 5 men in 96 days, or 1 man in 480 days.

Second important advantage we gain is in combining effect of two variables, number of men and days onto one, that is, work amount.

Third advantage we will see when we combine work of teams of different types of workers, for example, men and women. For men we will use, Mandays as work measure and for Women, Womandays.

While combining work of teams we can simply add the Work measures in mandays, boydays, womandays or whatever the case may be.

Problem example 1.

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?

Solution example 1.

$\text{Work } W = 288 \text{ mandays} = 144 \text{ womandays}$

Or, $2 \text{ mandays} = 1 \text{ womanday}$, a woman achieves more, 1 woman working is equivalent to 2 men working, this is the stage when we derive the crucial relationship between work rates of two types of workers.

When 12 men and 12 women work together, it is equivalent to, $(12+24) = 36$ men working together.

So 36 men will do 288 mandays work in $\displaystyle\frac{288}{36} = 8$ days.

It is easily possible to get the solution wholly mentally in this approach.

Still faster solution - Mathematical reasoning

Without going into the mandays or womandays calculations, just by examining the given data we discover the crucial fact that 1 woman works double that of a man.

Taking 12 men as one team, the team of 12 women will be equivalent to two such men’s teams. So if 12 men and 12 women work together it is same as three men’s team of 12 working together. As time and worker number are inversely proportional, 3 men’s teams will complete the work in, $\displaystyle\frac{24}{3}=8$ days.

This approach is wholly conceptual and based on deductive reasoning (in this case, mathematical reasoning, which is a subset of deductive reasoning). If you can use this path, it will give you the solution fastest, though it is not as easy to follow as it seems.

Problem example 2.

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?

Solution example 2

This is a Boundary condition problem in which we need to carefully detect the boundary, because no straightforward formula can produce the answer. By boundary we mean the last event point up to which a regular event goes on happening unchanged. Crossing over the last point, the nature of event changes. We will be clear about this boundary concept as we proceed.

1st 10 days, 40 men worked, work done = 400 mandays,

2nd 10 days, 35 men worked, work done = 350 mandays,

3rd 10 days, 30 men worked, work done = 300 mandays,

4th 10 days, 25 men worked, work done = 250 mandays, till now in 40 days 1300 mandays work is done, while total work is 1680 mandays. We can proceed safely to the 5th segment of 10 days. This step takes care of unwittingly overshooting the target.

5th 10 days, 20 workers worked, work done = 200 mandays, till now  total of 1500 mandays work done, work left is 180 mandays which will be worked upon by 15 men. We have to consider then the 6th segment of 10 days.

6th 10 days, 15 workers worked, work done = 150 mandays, a total of 1650 mandays work done till now, 30 mandays work left with number of workers 10. 

This 6th and last full segment of 10 days is the boundary in this problem. Till this boundary, worker teams, even in regularly reduced numbers, worked for the full segment period of 10 days. Crossing this boundary, the full period is broken and job is finished in 3 more days.

Answer: In 63 days the work will then be done.

Work measure in mandays is useful here. But solution is procedural, step by step and is slow.

Faster solution by mathematical reasoning

As 40 men working together all through take 42 days to complete the job, and in the second case, every 10 days worker force reduces by 5, we can safely test for 50 days work portion completion. This is called an Educated guess.

In MCQ based tests, such an estimate based on Educated guess, on which a trial will be made, is very useful.

In 50 days, as 5 persons are going off every 10 days, we have number of men working as 40, 35, 30, 25, and 20 over first five consecutive 10 day segments. Average number of men working over first 50 days will be then 30 and work portion completed $30\times{50}=1500 \text{ mandays}$.

We have used the concept of average of first 'n' natural numbers, where 'n' is odd. This is a basic number system concept. The middle number 30 is the average with which the number of days can safely be multiplied.

Total mandays work being $42\times{40}=1680 \text{ mandays}$, after 50 days, 180 mandays work will be left and 15 men will have to work for the full 6th segment of 10 days producing 150 mandays work. In 60 days then, $1500+150=1650 \text{ mandays}$ work will be finished.

Work left will be 30 mandays in the seventh 10 day segment when 10 men will be working.

In further 3 days, that is, in a total of 63 days the work will be finished.

No on paper calculation is necessary in this approach.

This speedy solution was possible because of the use of natural number concept.

More you use effective concepts from one topic area in solving a problem in another topic area, the faster you would be able to solve problems.

❮   Original version