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Set Theory Question

by datonman » Fri Jan 30, 2015 3:19 pm
The question is as follows:

In a certain class, 40% of the students have a pencil and 80% of the students have their books, calculate the minimum percentage of students having both a pencil and a book.


For this particular question, is it necessary to do a venn diagram only or is there a better way to get the answer?
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by MartyMurray » Fri Jan 30, 2015 3:35 pm
datonman wrote:The question is as follows:

In a certain class, 40% of the students have a pencil and 80% of the students have their books, calculate the minimum percentage of students having both a pencil and a book.


For this particular question, is it necessary to do a venn diagram only or is there a better way to get the answer?
Well, you can decide that, but another way to handle two overlapping groups or sets, A and B, is to use this formula.

Total = A + B - Both + Neither

In this case Total = 100 percent. Well call the percentage who have pencils A and the percentage who have books B.

So 100 = 40 + 80 - Both + Neither

40 + 80 = 120. So we are already over 100 by 20, even without considering any that may have neither a pencil nor their books.

So to get to 100, we need to subtract at least 20 and that's how many must have both.
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by [email protected] » Fri Jan 30, 2015 5:51 pm
Hi datonman,

You're asking about a math concept that's based in Overlapping Sets, but it does not require the use of the Overlapping Sets Formula nor the Tic-Tac-Toe Board to solve.

Here's a 'visual' way to think about this question:

40% have a pencil
80% have their books
We have to account for 100% of the students

The '0' symbol = 10% of the students

0000 = 40% with a pencil

00000000 = 80% have a book

IF....the 40% are "contained" in the 80%, then we have

0000
0000 40% of the students have BOTH
>>>>0000 40% of the students just have a book
>>>>>>>>00 the remaining 20% have neither

IF...we try to MINIMIZE the number who have BOTH, then we have to make it so as many of the students with a pencil DON'T have a book.

00000000 80% have a book
>>>>>>>>00 20% have just a pencil
>>>>>>00 The remaining 20% have BOTH

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Last edited by [email protected] on Sun Feb 01, 2015 1:10 pm, edited 1 time in total.
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by DavidG@VeritasPrep » Sun Feb 01, 2015 1:01 pm
Or we can use our fun little chart. Initially, we know that 40& have a pencil, which means that 60% don't have one. And we know that 80% have books, which means that 20% don't have them. We can represent that info like so:

Image


Next, if we want to minimize the number that have both, we'll want to maximize the number who just have Books but Not Pencils. Well, if 60 don't have pencils, and 80 have books, then 60 is the most that could have Books but not pencils. Now our chart looks like this:

Image


Now we just solve for the Pencils/Books cell. And we'll have:


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by Brent@GMATPrepNow » Sun Feb 01, 2015 1:06 pm
I thought I'd point out that David's "fun little chart" approach is also known as 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 baseball players, and the two characteristics are:
- has pencil or doesn't have pencil
- has books or doesn't have books

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 our free video: https://www.gmatprepnow.com/module/gmat- ... ems?id=919

Once you're familiar with this technique, you can attempt these additional practice questions:

Easy Problem Solving questions
- https://www.beatthegmat.com/the-aam-aadm ... 72242.html
- https://www.beatthegmat.com/finance-majo ... 67425.html

Medium Problem Solving questions
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- https://www.beatthegmat.com/prblem-solving-t279424.html

Difficult Problem Solving questions
- https://www.beatthegmat.com/ratio-problem-t268339.html
- https://www.beatthegmat.com/overlapping- ... 65223.html
- https://www.beatthegmat.com/fractions-t264254.html
- https://www.beatthegmat.com/overlapping- ... 64092.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-2

Easy Data Sufficiency questions
- https://www.beatthegmat.com/for-what-per ... 70596.html
- https://www.beatthegmat.com/ds-quest-t187706.html

Medium Data Sufficiency questions
- https://www.beatthegmat.com/sets-matrix-ds-t271914.html
- https://www.beatthegmat.com/each-of-peop ... 71375.html
- https://www.beatthegmat.com/a-manufacturer-t270331.html
- https://www.beatthegmat.com/in-costume-f ... 69355.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-1

Difficult Data Sufficiency questions
- https://www.beatthegmat.com/double-set-m ... 71423.html
- https://www.beatthegmat.com/sets-t269449.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-3

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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