How Many of the Students in a Certain Class are Taking Both a History and a Science Course? GMAT Data Sufficiency

Question: How Many of the Students in a Certain Class are Taking Both a History and a Science Course?

1.  Of all the students in the class, 50 are taking a history course
2.  Of all the students in the class, 70 are taking a science course

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“How Many of the Students in a Certain Class are Taking Both a History and a Science Course?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiencycomprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Approach Solution 1:

Given the prompt, before we get to the statements, think about everything we don't know. We don't know how many students are taking history, or are taking science, or are in the class as a whole.
It's probably apparent that neither statement by itself is sufficient to answer the prompt question. Let's consider them together.
Suppose we know that

1. 50 students are taking history
2. 70 students are taking science

and we want to know about the overlap, the folks taking both. We still don't know the class size.
Consider these two scenarios:
SCENARIO #1: Class size = 70
In this scenario, everyone is taking science, so the 50 who are taking history are necessarily taking it in addition to science. In this scenario, those 50 would be the overlap, the number of students taking both.
SCENARIO #2: Class size = 1000
An extraordinarily large class size, perhaps typical of a university. If we picked 50 people randomly and assigned them to history, then picked 70 people randomly and assigned them to science,it is likely to get a few overlaps. It would be easy for one set of 50 to take history, and other set of 70 to take science, leaving 880 students unenrolled in either. So, there may be no overlap, or there may be some handful of students enrolled in both. We have no way of determining.
One scenario led us to one answer. Another scenario left the prompt question unanswered. We would only have sufficient information if every imaginable scenario produced the same answer to the prompt question. Given the information we have, even with both students, we have absolutely no hope of answering the prompt question. Even with both statements combined, the information is insufficient.

Correct Answer: E

Approach Solution 2:

In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.
Total = history + Science - Both + Neither.
The class can have 4 sets of Students.

1. Take only history:
2. Take only Science:
3. Take both.
4. Take neither of the two.

1. WE have been given 50 but we haven't been told how many of them also do Science. Insuff
2. WE have been given 70 but we haven't been told how many of them also do history. Insuff.

St 1 + St 2 :-
We still don't know about total class size and we don't know value of Neither.

Correct Answer: E

Approach Solution 3:

Before we go on to the statements in light of the prompt, consider what we don't know. We have no idea how many kids are enrolled in science, history, or the class as a whole.
It should be clear that none of the statements alone is enough to respond to the question. Let's think about them all at once.
Say we already know this

There are 50 students taking history.
There are 70 kids taking science, and we want to know whether there are any students taking both. The size of the class is currently unknown.
Think about these two examples:

Example 1: A 70-person class
Everyone in this situation is taking science, hence the 50 individuals enrolled in history must also be taking science. In this instance, 50 students would be the overlap, or the number taking both courses.

Example 2: A 1000- students class 
a class size that is unusually large—possibly typical of a university. There would probably be some overlaps if we randomly chose 50 people to be allocated to history and then 70 people to be assigned to science. It would be simple for one group of 50 to join in history and another group of 70 in science, leaving 880 individuals without a placement in either course. Therefore, there might not be any overlap or there might be a small number of students taking both. There is no way for us to know.

We came at one solution from one circumstance. Another example failed to address the prompt question. Only if every possible situation resulted in the same response to the prompted question would we have enough data. Even with both students, we have no chance of being able to answer the prompted question with the facts we now have. Even when both claims are taken together, there is not enough data.

Correct Answer: E

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