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

Question

In a survey of 200 college graduates, 30 percent said they had received student loans during their college careers, and 40 percent said they had received scholarships. What percent of those surveyed said that they had received neither student loans nor scholarships during their college careers?

(1) 25 percent of those surveyed said that they had received scholarships but no loans. 
(2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.

Bunuel · Accepted Answer

In a survey of 200 college graduates, 30 percent said they had received student loans during their college careers, and 40 percent said they had received scholarships. What percent of those surveyed said that they had received neither student loans nor scholarships during their college careers?

30 percent received student loans --> 200*0.3 =  60  graduates received  loans ;
40 percent received scholarships --> 200*0.4 =  80  graduates received  scholarships .

200 = {loans} + {scholarships} - {both} + {neither} ;
 200 = 60 + 80 - {both} + {neither} ;
 {neither} = 60 + {both} .

Question:  {neither} = ?

As  {neither} = 60 + {both}  then we should calculate # of students who received both loans and scholarships.

(1) 25 percent of those surveyed said that they had received scholarships but no loans:

{scholarships} - {both} = 0.25*200 ;
 80 - {both} = 50 ;
 {both} = 80 - 50 = 30 ;
 {neither} = 60 + {both} = 60 + 30 = 90 .

Sufficient.

(2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships:

0.5*{loans} = {both} ;
 0.5*60 = 30 = {both} ;
 {neither} = 60 + {both} = 60 + 30 = 90 .

Sufficient.

Answer: D.

PrashantPonde · Answer

Based on the information given in the question stem we can put below information in the statistics table.

Attachment:

Students.jpg [ 99.12 KiB | Viewed 63851 times ]

We need to find "neither" i.e. "students with neither loans nor scholarships" - Marked in RED.
So if DS statement gives any information about the cells in YELLOW will lead you to calculated answer.

Statement(1)
SUFFICIENT: 25% (i.e. 50 students) are with scholarships but no loans. Based on this arrive at the answer "neither" = 90 (check the above image)

Statement(2)
SUFFICIENT: 50% of students with loans -> received scholarship i.e. 50% of 60 = 30 students. Based on this arrive at the answer "neither" = 90.

Hence choice(D) is the answer.

KarishmaB · Answer

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 Ques3.jpg 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)

JeffTargetTestPrep · Answer

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.

https://i.imgur.com/2D1gCby.png

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.

https://i.imgur.com/2Asb4FI.png

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.

https://i.imgur.com/JGUq5hS.png

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

kodaol · Answer

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"  ?

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

Thanks in advance!

KarishmaB · Answer

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?

BrentGMATPrepNow · Answer

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:
 https://s14.postimg.cc/46a5lax6l/schl1.jpg

In a survey of 200 college graduates...  
So, we'll add the population here:
 https://s14.postimg.cc/qw9aeage5/schl2.jpg

...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:
 https://s14.postimg.cc/qkru1izy5/schl3.jpg

...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:
 https://s14.postimg.cc/pmg2czbzx/schl4.jpg

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:
 https://s14.postimg.cc/5qk2xfuyl/schl5.jpg

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:
 https://s14.postimg.cc/4zrcrnsl9/schl6.jpg

Since the boxes in the bottom row must add to 140, we can determine the value that goes in the starred box:
 https://s14.postimg.cc/7bzcp956l/schl7.jpg 
 
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:
 https://s14.postimg.cc/5qk2xfuyl/schl5.jpg

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:
 https://s14.postimg.cc/bu1nhcl8d/schl8.jpg

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:
 https://s14.postimg.cc/dzvybuoot/schl9.jpg

Next, since the boxes in the bottom row must add to 140, we can determine the value that goes in the starred box:
 https://s14.postimg.cc/43uvc7iwt/schl10.jpg 
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:
 https://www.youtube.com/watch?v=jK-tiBrrf04

TheUltimateWinner · Answer

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__

Bunuel · Answer

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

lotif · Answer

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.

Bunuel · Answer

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.

GMATCoachBen · Answer

https://www.youtube.com/watch?v=DRRP3R24R7k

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.

Farhanashraf7 · Answer

Hey Bunuel, I am confused about statement 2 being sufficient here. The stem only says 50% of those with loans got scholarships but doesnt give the any data about how many among those who got scholarships took loans? Even that could contribute to 'both' right?

Bunuel · Answer

50 percent of those surveyed who said that they had  received loans  also said that they had  received scholarships   directly translates to 0.5*{loans} = {both}.

50% of those who  received loans  also  received scholarships , meaning 50% of the loans group had both loans and scholarships.

totaltestprepNick · Answer

D https://www.youtube.com/watch?v=fUFD2eTEn3k Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

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

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