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overlapping set: of 80 students in the eighth grade

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overlapping set: of 80 students in the eighth grade

by buoyant » Mon Jan 13, 2014 10:08 am
Of 80 students in the eighth grade, 35 played basketball and 19 made the Dean's List. How many of the students neither made the Dean's List, nor played basketball?

10 students played basketball and made the Dean's List
44 students played basketball or made the Dean's List or both

[spoiler]OA: D[/spoiler]

want to know various ways to deal with statement 2.
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by GMATGuruNY » Mon Jan 13, 2014 12:54 pm
buoyant wrote:Of 80 students in the eighth grade, 35 played basketball and 19 made the Dean's List. How many of the students neither made the Dean's List, nor played basketball?

10 students played basketball and made the Dean's List
44 students played basketball or made the Dean's List or both

[spoiler]OA: D[/spoiler]

want to know various ways to deal with statement 2.
We can use the following equation:

Total = Basketball + Dean's list - Both + Neither.

The big idea with overlapping groups is to SUBTRACT THE OVERLAP.
When we count everyone in Group 1 (basketball) and everyone in Group 2 (Dean's list), those in BOTH groups (students who can BOTH played basketball AND made the Dean's list) get counted twice.
So that we don't double-count the students who belong to both groups, we SUBTRACT THE OVERLAP from the total.

Statement 1: 10 students played basketball and made the Dean's List
In other words, both = 10.
Since total = 80, basketball = 35, Dean's list = 19, and both = 10, we get:
80 = 35 + 19 - 10 + N
N = 36.
SUFFICIENT.

Statement 2: 44 students played basketball or made the Dean's List or both
Of the 80 students, 44 played basketball, made the Dean's list, or both played basketball and made the Dean's list.
Implication:
The remaining students neither played basketball nor made the Dean's list:
N = 80-44 = 36.
SUFFICIENT.

The correct answer is D.
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by Brent@GMATPrepNow » Tue Jan 14, 2014 6:54 am
buoyant wrote:Of 80 students in the eighth grade, 35 played basketball and 19 made the Dean's List. How many of the students neither made the Dean's List, nor played basketball?

(1) 10 students played basketball and made the Dean's List
(2) 44 students played basketball or made the Dean's List or both
We can use the Double Matrix Method to solve this question. 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 80 students, and the two characteristics are:
- play basketball and don't play basketball
- on Dean's list and not on Dean's list.

So, we can set up our diagram as follows:
Image

To learn more about this technique, watch our free video: https://www.gmatprepnow.com/module/gmat- ... ems?id=919

Now let's continue....

Target question: How many of the students neither made the Dean's List, nor played basketball?

So, let's place a star in the box that need to find the value for.
Image

Given: 35 played basketball and 19 made the Dean's List.
We can add this information to our diagram as follows:
Image

As you can see, we don't yet have sufficient information to determine the value that goes in the starred box.

Statement 1: 10 students played basketball AND made the Dean's List
We can add that information to the diagram as follows:
Image

At this point, we have enough information to determine the value that goes in every box:
Image

So, as we can see, 36 students neither made the Dean's List, nor played basketball
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 44 students played basketball or made the Dean's List or both
This statement is referring to more than 1 box.
In fact, it's saying that each of the 44 students can be found in one of the 3 highlighted boxes below:
Image

This means that the remaining 36 students must be in the non-highlighted box:
Image

So, as we can see, 36 students neither made the Dean's List, nor played basketball
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

-------------------------------------------------------

Here are some additional practice questions that can be solved using the Double Matrix Method:
- https://www.beatthegmat.com/mba/2011/05/ ... question-1
- https://www.beatthegmat.com/mba/2011/05/ ... question-2
- https://www.beatthegmat.com/mba/2011/05/ ... question-3
- https://www.beatthegmat.com/ds-quest-t187706.html
- https://www.beatthegmat.com/overlapping- ... 83320.html
- https://www.beatthegmat.com/finance-majo ... 67425.html
- https://www.beatthegmat.com/ds-french-ja ... 22297.html
- https://www.beatthegmat.com/sets-t269449.html#692540
- https://www.beatthegmat.com/in-costume-f ... tml#692116

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by buoyant » Tue Jan 14, 2014 12:33 pm
GMATGuruNY wrote: We can use the following equation:

Total = Basketball + Dean's list - Both + Neither.

The big idea with overlapping groups is to SUBTRACT THE OVERLAP.
When we count everyone in Group 1 (basketball) and everyone in Group 2 (Dean's list), those in BOTH groups (students who can BOTH played basketball AND made the Dean's list) get counted twice.
So that we don't double-count the students who belong to both groups, we SUBTRACT THE OVERLAP from the total.

Statement 1: 10 students played basketball and made the Dean's List
In other words, both = 10.
Since total = 80, basketball = 35, Dean's list = 19, and both = 10, we get:
80 = 35 + 19 - 10 + N
N = 36.
SUFFICIENT.

Statement 2: 44 students played basketball or made the Dean's List or both
Of the 80 students, 44 played basketball, made the Dean's list, or both played basketball and made the Dean's list.
Implication:
The remaining students neither played basketball nor made the Dean's list:
N = 80-44 = 36.
SUFFICIENT.

The correct answer is D.

Hi Mitch,
In the above, can i write the equation as :

Only B + Only D + both B and D + neither = Total ?

If above is what you mean to say, then i get how the answer is derived from statement 2.

I assume that in the equation you have mentioned [Total = Basketball + Dean's list - Both + Neither], you mean that Basketball = only basket ball+ both basket ball and dean's list and that Dean's List= only Dean's list+ both basket ball and dean's list

Am i getting this right?
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by buoyant » Tue Jan 14, 2014 1:05 pm
Brent@GMATPrepNow wrote: Statement 2: 44 students played basketball or made the Dean's List or both
This statement is referring to more than 1 box.
In fact, it's saying that each of the 44 students can be found in one of the 3 highlighted boxes below:
Image

This means that the remaining 36 students must be in the non-highlighted box:
Image

So, as we can see, 36 students neither made the Dean's List, nor played basketball
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

-------------------------------------------------------
Hi Brent,

The pictorial explanation was helpful indeed.

Could you please respond to the below thread of set problem as well ?
I tried to solve using double matrix method, but could not get the answer.

https://www.beatthegmat.com/nationwide-p ... tml#704705
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by GMATGuruNY » Tue Jan 14, 2014 1:30 pm
buoyant wrote:
GMATGuruNY wrote: We can use the following equation:

Total = Basketball + Dean's list - Both + Neither.

The big idea with overlapping groups is to SUBTRACT THE OVERLAP.
When we count everyone in Group 1 (basketball) and everyone in Group 2 (Dean's list), those in BOTH groups (students who can BOTH played basketball AND made the Dean's list) get counted twice.
So that we don't double-count the students who belong to both groups, we SUBTRACT THE OVERLAP from the total.

Statement 1: 10 students played basketball and made the Dean's List
In other words, both = 10.
Since total = 80, basketball = 35, Dean's list = 19, and both = 10, we get:
80 = 35 + 19 - 10 + N
N = 36.
SUFFICIENT.

Statement 2: 44 students played basketball or made the Dean's List or both
Of the 80 students, 44 played basketball, made the Dean's list, or both played basketball and made the Dean's list.
Implication:
The remaining students neither played basketball nor made the Dean's list:
N = 80-44 = 36.
SUFFICIENT.

The correct answer is D.

Hi Mitch,
In the above, can i write the equation as :

Only B + Only D + both B and D + neither = Total ?

If above is what you mean to say, then i get how the answer is derived from statement 2.

I assume that in the equation you have mentioned [Total = Basketball + Dean's list - Both + Neither], you mean that Basketball = only basket ball+ both basket ball and dean's list and that Dean's List= only Dean's list+ both basket ball and dean's list

Am i getting this right?
Given two groups B and D, each student has 2 options:
Option X: he/she is in NEITHER group
Option Y: he/she is in ONE OR BOTH groups

Here are two equations that can be used to count the students:
(Everyone in B) + (Everyone in D) - (Both B and D) + Neither = Total.
(Only B) + (Only D) + (Both B and D) + Neither = Total.
In these equations, the portions in red represent two different ways to account for all of the students in option Y.

Statement 2 indicates that the number of students in Option Y = 44.
Thus, regardless of which equation we use, the result is the same:
44 + N = 80
N = 36.
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