Analytical approach is essential for quickly simplifying a larger problem
The GATE level work time problem taken up is quite long in terms of number of words. It resembles more to a real life work place problem than a purely academic one because of a few hidden objectives that are managerial in nature.
Because of the long description of the problem and the partly hidden objectives, the problem may seem to be confusing to students used to conventional way of academic problem solving using mathematical deductions.
We will go ahead in explaining the analytical thinking when breaking up the problem towards solution, but in summary the problem is solvable in mind in about half a minute with the right analytical problem solving approach.
Wall painting is the profession of Nandini. She can paint a 10 ft by 10 ft wall in 2 days, whereas her hired painter Sonu takes 3 days to paint the same wall. Recently in a contract job she has promised to complete the painting of the walls and ceiling of a 10 ft by 15 ft room 10 ft high in 9 days. She pays Sonu on per day basis and being on business does not want to use his services more than necessary. To complete the contract job in time, for how many days would she engage Sonu?
- 7 days
- 5 days
- 3 days
- 6 days
Problem definition: first stage of picking up and defining numerical facts
In this long-worded problem the first task is to pick up the given numerical facts quickly. Once you isolate the numerical facts, you tend to get meaning of the facts. The facts given are:
- Work rate of Nandini is 100 sq ft in 2 days. Because of convenience we convert it immediately to 50 sq ft in a day.
- Work rate of hired help Sonu is 100 sq ft in 3 days. We don't convert this to per day work rate because of inconvenience of converting.
- Work amount of the contract is two walls each 100 sq ft, two walls each 150 sq ft and the ceiling 150 sq ft. We don't calculate the total amount and instead clearly specify the work in terms of work units. The total work consists of 5 work units, two 100 sq ft each and three 150 sq ft each. In a real life work where more than one worker is involved and working together is inconvenient, it is a practical approach to deal in terms of work units of well defined work amounts. When the work allocation time comes, it is easier to allocate the work units to individual workers according to their work capacity.
- Target time: Work is to be finished in a total of 9 days.
Problem definition: critical task of defining primary objective or target
Completing the job in 9 days seems to be the primary objective. But Nandini won't like to employ Sonu for more number of days than necessary. In other words, Nandini would always like to employ Sonu for minimum number of days possible. Thus in this real life work place problem, the primary objective is not just completing the job in time, but also completing it at minimum cost, that is, employing Sonu for minimum number of days.
To precisely define the primary objective of solution,
Nandini needs to engage Sonu for minimum number of days necessary for meeting the target of 9 days.
Problem solution: taking the steps to ideally meet the objective
The aspect of minimum number of days' engagement of Sony lies at the heart of the problem here.
How to achieve this objective when work capacity of both Sonu and Nandini herself are known?
The clear answer to this second question is,
Nandini herself must work for full 9 days herself.
This is elementary logical deduction we carry out frequently in problem situations using no mathematics.
As Nandini knows the total work requirement and work capacity of herself, it would be easy for her to get the amount of work she can do in 9 days, and how much work will be left.
Knowing this work shortfall and work capacity of Sonu, she can then easily calculate the minimum number of days for which she needs to hire Sonu.
Problem solution: attention to convenience of work
As this problem resembles a real life work place problem, it would be best if one or two complete wall or ceiling painting is allotted to Sonu. Imagine how inconvenient it would be to paint one wall, a portion by Nandini and remaining by Sonu.
So what does Nandini do?
The first thing she does is to mentally take up painting four walls plus ceiling, a five unit painting herself, estimating how many units she can finish in 9 days.
With the given dimensions, a pair of walls are each of area 100 sq ft, a second pair each of area 150 sq ft and the ceiling 150 sq ft.
With ability to paint 100 sq ft in 2 days, in 9 days she can then paint the three units of pair of 150 sq ft walls and the ceiling, a total of 450 sq ft, leaving 200 sq ft of two walls that Sonu will have to be engaged for 6 days to paint.
This part is the optimum choice of work units so that primary target as well as work convenience objectives are fulfilled.
With Nandini painting the pair of walls 150 sq ft each and the ceiling also 150 sq ft in 9 days, she will engage Sonu for minimum number of 6 days to paint the remaining pair of walls 100 sq ft each. The primary objective and secondary work convenience objectives fulfilled in this optimal allocation of work will be,
- Primary objective: The work will be completed in 9 days with Sonu working for minimum number of 6 days.
- Secondary work convenience objective: Nandini and Sonu can work independently on their set of walls or ceiling.
- Secondary work convenience objective: Nandini can call Sonu for any six days of work within the period of 9 days total work. It introduces a new dimension of flexibility in carrying out the work optimally.
- Secondary optimal resource utilization objective: Sonu will have to work full time for all six days. His paid services will be fully utilized.
This problem is by no means a workers working together problem, and analytical objective oriented approach not only hands over the 6 days solution, it also defines optimal work allocation in the process.
In other words, this is NOT a problem to approach using conventional mathematics. Optimal solution is reached quickly without any writing, just by thinking analytically step by step.
Summing up analytical step by step problem solving
Step 1: Assessment and required transformation of numerical given facts: Routine but important in any practical work place situation or solving any academic word problem.
Step 2: Forming primary objective of fulfilling the target and employing Sonu for minimum number of days: At this stage, no numbers are involved, just objectives.
Step 3. Forming first conclusion that Nandini has to work for full 9 days to employ Sonu for minimum number of days: This conclusion is arrived by logical deduction based on the primary objective.
Step 4: Nandini selects for herself optimum combination of work units that she will take 9 days to complete. This work unit break up concept introduces convenience of work as well as ease of reaching the solution.
Step 5: The work left out determines for how many minimum number of days Sonu needs to be engaged. In the process a number of other real life work place convenience objectives are fulfilled that are unspoken in the problem description, but very real in practical environment.
End note: The first time you encounter such a problem, it might take you more time to solve or you may not adopt the analytical approach. But that is your first time. When you go into your final exam, by then you have gone through enough preparations by solving sufficient number of problems with an analytical approach so that you would be able to proceed easily to the optimal solution usually in your mind in a very short time.
Our recommendation: Go through the above process of solution more than once to understand and absorb the concepts fully. These are simple and easy to understand.
Useful resources to refer to
Guidelines, Tutorials and Quick methods to solve Work Time problems
How to solve a GATE level long Work Time problem analytically in a few steps 1