How to identify the approach for Overlapping set Questions?

Hi,

I can understand this fairly common problem in identifying the best approach to solve a particular set problem i.e. whether to use Grid method or venn diagram.

Let me begin by saying that both of these methods work correctly i.e. you can apply either of these to any problem. However, as you said, grid approach is better than venn diagram approach in some questions and venn diagram approach is better than grid approach in others. How do we find out this when we look at a question? Let's try to find this by using examples:

Let's begin with the first question you asked:

1. At a certain hospital, 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts. At the same time, 70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness. If 80% of the interns receive fewer than 6 hours of sleep, what percent of the interns report no feelings of tiredness during their shifts?

For this question, Karishma, in the above post, said that many people find grid approach better to solve this question. This is true; not only grid approach seems easier in this case, venn diagram looks very difficult to use.

Now, consider a variation of the above question:
2. At a certain hospital, 75% of the interns are both tall and smart. At the same time, 70% of the interns who are not tall, are not smart. If 80% of interns are tall, what percent of interns are not smart? (Here, I have just denoted "people who receive fewer than 6 hours of sleep" as tall people and "people who report feeling tired during the shifts" as smart people. The number of people in each set remains same)

Here, both venn and grid method become equally easy (or difficult),due to the changes made in the way sets' are described. I hope you are able to see the difference.

Lets see another variation:
3. At a certain hospital, 75% of interns receive fewer than 6 hours of sleep and 70% of interns report feeling tired. If 60% of interns receive fewer than 6 hours of sleep and report tiredness, what is the percentage of interns who receive more than 6 hours of sleep and don't report feeling tired? (This question changes the information presented without changing the meaning of the sets)

Here, venn diagram is easier to than grid method (here sets' description remains same; however data is changed). Are you able to see?

Another variation:
4. At a certain hospital, 75% people are tall and 70% people are smart. If 60% are both tall and smart, what percentage is neither tall nor smart? (This is a modification of question 3. Here, I have just denoted "people who receive fewer than 6 hours of sleep" as tall people and "people who report feeling tired during the shifts" as smart people)

Here, venn diagram is the most intuitive method to use.

So, What has happened?

In the first variation, grid method was the easier way; in second variation, both venn and grid method looked equally easy; in third variation, venn diagrams became the easier method; and in the fourth variation, venn diagram became the most intuitive method. Why so?

The understanding of this goes back to the way we think. I have basically played with two variables here:

1. First variable is the complexity in the description of set. If the description of the sets are complex, it is easier to deal in grids than venn diagrams. For example - as we move from question 2 to question 1, grid method becomes much easier to implement than venn diagram. Similar effect is felt as we move from question 4 to question 3.

2. Second variable is how much information is given in conjunctive form (A ∩ B, A' ∩ B, A ∩ B', A' ∩ B'). The more the information is given in this format, easier it becomes to use grid method. This is because all the conjunctives except A ∩ B, are not intuitive in venn diagrams (complements are difficult to deal/visualise in venn diagrams), whereas they seem so easy to use when dealing with grids. This is because each of the grid box is very clearly a conjunction, one of these four combinations. So, as we find these conjunction given in the question, we easily fill up our grid. For example - as we move from Q3 to Q1 or Q4 to Q2, we find that grids become easier to implement and venn diagram becomes difficult.

So, if sets' descriptions are complex (phrases rather than words) and if most of the information is given in conjunctive form, then it is best to use grid method.
And if sets' descriptions are easy and information is given mainly in non-conjunctive form (e.g. Q3 and Q4), then it is best to use venn diagram method.
Other than these two cases, you need to take call depending on its closeness to the above two cases.

Regarding your other two questions:

Each light bulb at Hotel California is either incandescent or fluorescent. At a certain moment, forty percent of the incandescent bulbs are switched on, and ten percent of the fluorescent bulbs are switched off. If eighty percent of all the bulbs are switched on at this moment, what percent of the bulbs that are switched on are incandescent?
Grid method is preferable since the questions provide data on both incandescent (suppose set A) and fluorescent (set A') as well as both switched on (Set B) and switched off (Set B'). As we said, in venn diagrams, it is usually more complex to deal with complement of sets, venn diagram is not recommended for this.

Q3) 100 people are attending a newspaper conference. 45 of them are writers and more than 38 are editors. Of the people at the conference, x are both writers and editors and 2x are neither. What is the largest possible number of people who are both writers and editors?
Here, you can easily see, based on the two parameters, venn diagram should be easier.

It might seem difficult to you to use this logic of choosing the right method for the first few days, however, with time, this logic will become embedded in your intuition and you'll be able to decide the best method just by looking cursorily over the question. Believe me this is true. This choice between methods is so intuitive to me that it took me 3-4 hours to figure out logic behind my intuition and answer your question.

I know I haven't solved any of the questions above but my post is already very long, it would become unmanageable if I add the solutions also. Besides, I think you should be able to solve them using the right methods.

If you have any further queries, please feel free to ask.

Cheers,
CJ