In a survey of 200 college graduates, 30 percent said they

In my opinion, Venn diagrams are most effective in solving such questions.

Students who received loans = 30% of 200 = 60
Students who received schol = 40% of 200 = 80


There are 3 regions in the figure - red identifying people who received only loans. Yellow for people who received only scholarships and orange for those who received both. To get neither, i.e. the white region, we need to know how many receive both - the orange region.

200 - Neither = 60 + 80 - Both

(1) 25 percent of those surveyed said that they had received scholarships but no loans.
This tells us that the yellow region is 50. This means the orange region is 30 since the entire circle is 80. This gives us both and hence is enough to answer the question.

(2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.
50% of people who received loans received scholarships too. Since 60 people received loans, 30 received scholarships too. This means orange region is 30 i.e. both is 30. This statement alone is also sufficient to answer the question.

Answer (D)

Solution:

The easiest way to solve this problem is to set up a double set matrix. In our matrix we have two main categories: student loans and scholarships. More specifically, our table will be labeled with:

1) Received student loans (Loans)

2) Did not receive student loans (No Loans)

3) Received a Scholarship (Scholarship)

4) Did not receive a scholarship (No Scholarship)

(To save room on our table headings, we will use the abbreviations for these categories)

We are given that there are a total of 200 college graduates in the survey. We also are given that 30 percent of those graduates received student loans and 40 percent received scholarships.

Thus,

200 x 0.3 = 60 received student loans

200 x 0.4 = 80 received scholarships

We are trying to determine what percent of those surveyed said that they had received neither student loans nor scholarships.

Let’s fill all this information into a table. Note that each row sums to create a row total, and each column sums to create a column total. These totals also sum to give us the grand total, designated by 200 at the bottom right of the table.



Statement One Alone:

25 percent of those surveyed said that they had received scholarships but no loans.

Using statement one we can determine the number of students who received scholarships but no loans.

200 x 0.25 = 50 students who received scholarships but no loans.

We can fill the above information into our table.



Thus, the percent of those surveyed who said that they had received neither student loans nor scholarships is (90/200) x 100 = 45%. Statement one is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

50 percent of those surveyed who said that they had received loans also said that they had received scholarships.

We are given that 50 percent of those surveyed who said they had received loans also said that they had received scholarships. From the given information we know that 60 students received loans; thus, we can determine the number of these 60 students who also received scholarships.

60 x 0.5 = 30 students who received loans who also received scholarships

We can fill the above information into our table.



Thus, the percent of those surveyed who said that they had received neither student loans nor scholarships is (90/200) x 100 = 45%. Statement two is sufficient to answer the question.

The answer is D

The statement (2) confuses me, it seems like it is not exhaustive to assume the total number of those who received both.
Could it not be, that "50 percent of those surveyed who said that they had received loans also said that they had received scholarships" [... and x% of those surveyed who said that they had received scholarships also said that they had received loans.]?

Could you please correct my error in thinking and explain why the above statement cannot be a case?

Thanks in advance!

Stmnt 2 does mean "50 percent of those surveyed who said that they had received loans also said that they had received scholarships" and it is exhaustive to get the total number of those who received both.

Note that when we talk about overlap of two sets, say A and B, it is enough to say that 50% of A is B too to get Both. Say A has 10 elements. 5 of those are B too. These 5 will be the only ones in A as well as B. The other 5 in A are not in B. A includes all elements that are A. So there can be no other element besides these 10 that are A. So saying that 20% of B are A too would be the same 5 elements only. Since they are A too, they MUST be in the set A. Hence, only 5 will be in Both.

Does that help?

We can solve this question using the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of college graduates, and the two characteristics are:
- received scholarships or didn't receive scholarships
- received loans or didn't receive loans

So, we can set up our diagram as follows:


In a survey of 200 college graduates...
So, we'll add the population here:


...30 percent said they had received student loans
30% of 200 = 60, so 60 students received loans, which also means 140 students received no loans.
Add this to our diagram:


...and 40 percent said they had received scholarships
40% of 200 = 80, so 80 students received scholarships, which also means 120 students received no scholarships.
Add this to our diagram:


Target question: What percent of those surveyed said that they had received neither student loans nor scholarships during their college careers?
Let's place a star in the box that represents this portion of the population to remind us of our goal:


Statement 1: 25 percent of those surveyed said that they had received scholarships but no loans
25% of 200 = 50, so 50 students can be placed in the following box:


Since the boxes in the bottom row must add to 140, we can determine the value that goes in the starred box:


So, 90 students received neither student loans nor scholarships.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

At this point, we'll revert back to the diagram we created with the given information:


Statement 2: 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.
Our diagram tells us that 60 students received loans.
50% of 60 = 30, so 30 students received loans AND scholarships
We can place this information as follows:


Since the boxes in the left-hand column row must add to 80, we can determine the value that goes in the bottom-left box:


Next, since the boxes in the bottom row must add to 140, we can determine the value that goes in the starred box:

So, 90 students received neither student loans nor scholarships.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

This question type is VERY COMMON on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video:

Hello Bunuel
Thanks for the nice explanation. +1 for you.
It seems that the question asked about percentage of "neither student loans nor scholarships" NOT the core number of "neither student loans nor scholarships". So, the percent should be 45% (\(\frac{90X100}{200}=45\))
Am I missing anything?
Thanks__

Yes, but knowing {neither} is enough to get the percentage and we can stop there.

Why is it not possible to use the general formulas

A + B + (A and B) + Neither = total

Or

Only A + Only B - (A and B) + Neither = total

For this question? I came to a dead end using these, whereas Venn or Matrix works quite easily.

Because the above is not correct. It should be:

Total = A + B – Both + Neither
Total = Only A + Only B + Both + Neither

You can check how you can apply correct formulae to solve this question HERE.

19. Overlapping Sets

• Theory
  Overlapping Sets Made Easy!
  How to draw a Venn Diagram for problems
  ADVANCED OVERLAPPING SETS PROBLEMS
  Formulae for 3 overlapping sets
  
  • Questions
    The E-GMAT Sets Triad: 3 Exciting Sets Questions!
    The Word “Or” in GMAT Math
    DS Questions
    PS Questions

For more:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.

Apologies, that's what I meant. Got the signs wrong for some reason. But my point still stands re this question isnt best answered via the formula. At least I couldnt, despite the question seeming to me to allow for that method. But with matrix or venn it was simple. Point of learning for myself that some questions are better answered with venn or matrix.

3 Key Habits:

1) Quickly recognize that we have two categories that are Yes-No (loans and scholarships), so we can quickly set up a Matrix for this Overlapping Sets problem.

2) We can ignore the "200" number, because everything else is in "percent". We just use 100% for the total.

3) If we know two numbers in a row or column, we always know the third number, without having to calculate.