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.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
OA is D. Please show the steps to resolve this problem.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
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survey of 200 college grads
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punit.kaur.mba
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Yes, Answer is (D).
Here is the explaination -
Notations : -
(Total surveyed students) - T
Students with either with Loan or Scholarship - L U S(L union S which is mathematical notation in Set Theory)
Students with Loan and Scholarship - L Intersection S - L & S
Students with neither - Negation(L union S) --> ~(L U S)
Given : -
Universal Set = Students who have either Loan or Scholarship + Students who have neither
200 = (L U S) + ~(L U S)
L U S = Students with Loans ( L )+ Students with Scholarships (S ) - ( L & S) [ who have both]
Given - 30 percent said they had received student loans during their college careers
Hence 30% of 200 = 60---> L ( Note : some of them might have scholarships)
40 percent said they had received scholarships
Hence 40% of 200 = 80 ---> S ( Note: some of them have Loans)
So, L U S = 60 + 80 - (L&S)
Stmt1:25 percent of those surveyed said that they had received scholarships but no loans
From given S = 80
S only( excluding people with Loans) = 25 % of 200 = 50
Hence, People with S & L = 80 - 50 = 30
Hence,
L U S = 60 + 80 - 30 = 110
~(LUS) = T - (L U S)
200 - 110 = 90
Stmt1 is sufficient alone
Stmt2 - 50 percent of those surveyed who said that they had received loans also said that they had received scholarships
50 % of 60 = 30 ---> L & S
Same way of solving as above,
Stmt 2 is Sufficient
Hence Answer is (D)
Here is the explaination -
Notations : -
(Total surveyed students) - T
Students with either with Loan or Scholarship - L U S(L union S which is mathematical notation in Set Theory)
Students with Loan and Scholarship - L Intersection S - L & S
Students with neither - Negation(L union S) --> ~(L U S)
Given : -
Universal Set = Students who have either Loan or Scholarship + Students who have neither
200 = (L U S) + ~(L U S)
L U S = Students with Loans ( L )+ Students with Scholarships (S ) - ( L & S) [ who have both]
Given - 30 percent said they had received student loans during their college careers
Hence 30% of 200 = 60---> L ( Note : some of them might have scholarships)
40 percent said they had received scholarships
Hence 40% of 200 = 80 ---> S ( Note: some of them have Loans)
So, L U S = 60 + 80 - (L&S)
Stmt1:25 percent of those surveyed said that they had received scholarships but no loans
From given S = 80
S only( excluding people with Loans) = 25 % of 200 = 50
Hence, People with S & L = 80 - 50 = 30
Hence,
L U S = 60 + 80 - 30 = 110
~(LUS) = T - (L U S)
200 - 110 = 90
Stmt1 is sufficient alone
Stmt2 - 50 percent of those surveyed who said that they had received loans also said that they had received scholarships
50 % of 60 = 30 ---> L & S
Same way of solving as above,
Stmt 2 is Sufficient
Hence Answer is (D)
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punit.kaur.mba
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I agree solving the problem the way I did would be a little time consuming . But somehow, I am confident of the answer when I do it my usual way.
I haven't gone through any of Quant material or strategies yet. If I come across anything good which will save me time, I'll definitely go for tht.
Which Mgmat method r u talking about? What is the book u r using for quant?
I haven't gone through any of Quant material or strategies yet. If I come across anything good which will save me time, I'll definitely go for tht.
Which Mgmat method r u talking about? What is the book u r using for quant?
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akshatsingh
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This can certainly be solved by Venn Diagrams but a faster method would be to use the charting method suggested by Manhattan. But, I have a doubt with charting here 
Can Ne1 suggest where am I goin wrong? Here is the chart.
S=>NO. OF STUDENTS WHO RECEIVED SCHOLARSHIPS.
NS=>NO. OF STUDENTS WHO DID NOT RECEIVE SCHOLARSHIPS.
L=>NO. OF STUDENTS WHO RECEIVED LOANS
NL=>NO. OF STUDENTS WHO DID NOT RECEIVE LOANS
T=>TOTAL
30% = 60
40% = 80
-----S -------NS-----T
L-------------------- 60(GIVEN)
NL------------?------140
T----80------120----200
(GIVEN)
Stmt 1 =>
25% = 50
-----S -------NS--------T
L----30-----------------60(GIVEN)
NL---50------90--------140
T----80-------120------ 200
(GIVEN)
sufficient since this gives me the answer (but watch the no. of students with both loans and scholarships =30 get conflicted by stmt 2)
Stmt 2 = > students with S&L 50% = 100 (which opposes stmt1). Why is this so? Its firmly believed that two stmts cant oppose each other.
Additionally, the total comes to 80 then how can s&l be 100? I think I am making a mistake somewhere.Can ne plz help?
Can Ne1 suggest where am I goin wrong? Here is the chart.S=>NO. OF STUDENTS WHO RECEIVED SCHOLARSHIPS.
NS=>NO. OF STUDENTS WHO DID NOT RECEIVE SCHOLARSHIPS.
L=>NO. OF STUDENTS WHO RECEIVED LOANS
NL=>NO. OF STUDENTS WHO DID NOT RECEIVE LOANS
T=>TOTAL
30% = 60
40% = 80
-----S -------NS-----T
L-------------------- 60(GIVEN)
NL------------?------140
T----80------120----200
(GIVEN)
Stmt 1 =>
25% = 50
-----S -------NS--------T
L----30-----------------60(GIVEN)
NL---50------90--------140
T----80-------120------ 200
(GIVEN)
sufficient since this gives me the answer (but watch the no. of students with both loans and scholarships =30 get conflicted by stmt 2)
Stmt 2 = > students with S&L 50% = 100 (which opposes stmt1). Why is this so? Its firmly believed that two stmts cant oppose each other.
Additionally, the total comes to 80 then how can s&l be 100? I think I am making a mistake somewhere.Can ne plz help?
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cubicle_bound_misfit
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Stacey Koprince
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Received a PM asking me to respond, specifically in terms of how to use the Double-Set Matrix (the MGMAT grid method) to solve the problem.
It's a little tough graphically to show the grid here, so bear with me if the formatting looks funny.
First set: loan vs. no loan
Second set: scholarship vs. no scholarship
Find: no loan and no scholarship
.............S.........NS.........Tot
L
NL.....................??
Tot...............................200
30% = L. 30% of 200 = 60.
.............S.........NS.........Tot
L...................................60
NL.....................??........200-60
Tot...............................200
40% = S. 40% of 200 = 80
.............S.........NS.........Tot
L...................................60
NL.....................??.........140
Tot........80.......200-80.....200
Statement 1:
25% of total = S and NL
25% of 200 = 50
.............S.........NS.........Tot
L...................................60
NL.........50..........??........140
Tot........80..........120......200
Can solve for NL + NS. Sufficient.
Statement 2:
50% of L's also had S's. [Note: NOT 50% of the total 200! Go read the sentence again. The word "who" is the key - it changes the meaning and tells you we're taking a percentage of a subset, not the overall group!]
L's = 60, so 50% of 60 = 30.
.............S.........NS.........Tot
L...........30......................60
NL.....................??.........140
Tot........80.......200-80.....200
Can solve for NL + NS. Sufficient.
It's a little tough graphically to show the grid here, so bear with me if the formatting looks funny.
First set: loan vs. no loan
Second set: scholarship vs. no scholarship
Find: no loan and no scholarship
.............S.........NS.........Tot
L
NL.....................??
Tot...............................200
30% = L. 30% of 200 = 60.
.............S.........NS.........Tot
L...................................60
NL.....................??........200-60
Tot...............................200
40% = S. 40% of 200 = 80
.............S.........NS.........Tot
L...................................60
NL.....................??.........140
Tot........80.......200-80.....200
Statement 1:
25% of total = S and NL
25% of 200 = 50
.............S.........NS.........Tot
L...................................60
NL.........50..........??........140
Tot........80..........120......200
Can solve for NL + NS. Sufficient.
Statement 2:
50% of L's also had S's. [Note: NOT 50% of the total 200! Go read the sentence again. The word "who" is the key - it changes the meaning and tells you we're taking a percentage of a subset, not the overall group!]
L's = 60, so 50% of 60 = 30.
.............S.........NS.........Tot
L...........30......................60
NL.....................??.........140
Tot........80.......200-80.....200
Can solve for NL + NS. Sufficient.
Please note: I do not use the Private Messaging system! I will not see any PMs that you send to me!!
Stacey Koprince
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Manhattan GMAT
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Stacey Koprince
GMAT Instructor
Director of Online Community
Manhattan GMAT
Contributor to Beat The GMAT!
Learn more about me
Thanks for explaining that Stacey. I'm just not sure about one part. The question says 50% of students who received loans also got scholarships. Is this the total number of students who received both loans and scholarships? Wouldn't they also need to have the percentage of scholarship receiving students who also received loans?
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ikaplan
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For these kind of problems, I always use the grid- the results are evident- thanks to the grid, 99% of my answers are correct.
I find the grid equally useful both for PS and DS questions. Once you master it, the grid is in your blood
I find the grid equally useful both for PS and DS questions. Once you master it, the grid is in your blood
"Commitment is more than just wishing for the right conditions. Commitment is working with what you have."
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vijaykondepudi
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I see there is a question in this thread from malman, which is not answered:
Statement 2 says:
(2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.
50 % of those that received scholarships also received loans. If I assume that the overlap set consists of
Case 1> People from those who received Scholarships who also received Loans
+
Case 2> People from those who received Loans, who also received Schorlarships
Statement II, addresses only the Case 1, with no information about case 2.
How can we say it's sufficient ?
Thank You !!
I had a similar doubt regarding Statement II.Thanks for explaining that Stacey. I'm just not sure about one part. The question says 50% of students who received loans also got scholarships. Is this the total number of students who received both loans and scholarships? Wouldn't they also need to have the percentage of scholarship receiving students who also received loans?
Statement 2 says:
(2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.
50 % of those that received scholarships also received loans. If I assume that the overlap set consists of
Case 1> People from those who received Scholarships who also received Loans
+
Case 2> People from those who received Loans, who also received Schorlarships
Statement II, addresses only the Case 1, with no information about case 2.
How can we say it's sufficient ?
Thank You !!
-
santhosh_katkurwar
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I understand we can solve this by venn diagram or grid but Can someone also solve this formula also?
Total = Group 1 + group 2 - both + neither
200 = 60 +80 - both + neither
Now I am not able to go ahead with the fact statements. Please help.
Total = Group 1 + group 2 - both + neither
200 = 60 +80 - both + neither
Now I am not able to go ahead with the fact statements. Please help.
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GMATGuruNY
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It is more efficient to plug percent values into the group equation rather than actual numbers, letting the total = 100.santhosh_katkurwar wrote:I understand we can solve this by venn diagram or grid but Can someone also solve this formula also?
Total = Group 1 + group 2 - both + neither
200 = 60 +80 - both + neither
Plugging the percents for the two groups into the group equation, we get: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?
100 = 30 + 40 - both + neither.
Since 40% receive a scholarship -- and 25% receive ONLY a scholarship but NOT a loan -- the percent who receive BOTH a scholarship AND a loan = 40-25 = 15.(1) 25 percent of those surveyed said that they had received scholarships but no loans.
Plugging both = 15 into the blue equation above, we get:
100 = 30 + 40 - 15 + neither
100 = 55 + neither
45 = neither.
Since 45% of the 200 students receive neither a loan nor a scholarship, the number who receive neither = (45/100)(200) = 90.
SUFFICIENT.
Since 1/2 of the 30% who receive a loan also receive a scholarship, the percent who receive BOTH a loan AND a scholarship = (1/2)(30%) = 15.(2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.
Plugging both = 15 into the blue equation above, we get:
100 = 30 + 40 - 15 + neither
100 = 55 + neither
45 = neither.
Since 45% of the 200 students receive neither a loan nor a scholarship, the number who receive neither = (45/100)(200) = 90.
SUFFICIENT.
The correct answer is D.
Solving with a double-matrix seems easier.
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As a tutor, I don't simply teach you how I would approach problems.
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