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

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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
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udaymathapati wrote:
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.

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

Attachment:

Ques3.jpg [ 13.55 KiB | Viewed 50062 times ]

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.

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.

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M8 wrote:
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.

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:

2) Did not receive student loans (No Loans)

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:

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.

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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
Bunuel wrote:
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?

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.

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

Sufficient.

The statement (2) confuses me, it seems like it is not exhaustive to assume the total number of those who received both.

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

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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
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kodaol wrote:
Bunuel wrote:
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?

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.

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

Sufficient.

The statement (2) confuses me, it seems like it is not exhaustive to assume the total number of those who received both.

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

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?
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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
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udaymathapati wrote:
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.

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:

So, we can set up our diagram as follows:

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

30% of 200 = 60, so 60 students received loans, which also means 140 students received no loans.

40% of 200 = 80, so 80 students received scholarships, which also means 120 students received no scholarships.

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:

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

Cheers,
Brent

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

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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
Bunuel wrote:
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?

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.

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

Sufficient.

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__
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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
Bunuel wrote:
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?

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.

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

Sufficient.

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.
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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
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.
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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
rezalotif wrote:
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

For more:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
Bunuel wrote:
rezalotif wrote:
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

For more:
ALL YOU NEED FOR QUANT ! ! !

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.
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In a survey of 200 college graduates, 30 percent said they [#permalink]
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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.
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Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
how do we know then there are no students who chose not to answer, I marked E due that else was an easy question. Is there something I am missing in the language ?
Re: In a survey of 200 college graduates, 30 percent said they [#permalink]
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