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Overlapping Sets

This topic has 2 expert replies and 2 member replies
jose.mario.amaya Junior | Next Rank: 30 Posts Default Avatar
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Overlapping Sets

Post Wed Jun 12, 2013 9:08 am
At company X, 44% of the employees that own laptops do not own cellphone, and 30% of the employees that own cellphone do not own laptops. If the number of employees without laptops is 40% less than the employees with laptops, then what percent of the company's employees neither own a laptop nor a cellphone ??

A. 17.5
B. 22.5
C. 24
D. 26
E. 28

B

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GMAT/MBA Expert

Post Wed Jun 12, 2013 9:57 am
jose.mario.amaya wrote:
At company X, 44% of the employees that own laptops do not own cellphone, and 30% of the employees that own cellphone do not own laptops. If the number of employees without laptops is 40% less than the employees with laptops, then what percent of the company's employees neither own a laptop nor a cellphone ??

A. 17.5
B. 22.5
C. 24
D. 26
E. 28

B
This is a great candidate for a technique called the Double Matrix Method. It can be used for most questions featuring a population in which each member has two criteria associated with it.
Here, the criteria are:
- own laptop (L) and don't own laptop (~L)
- own cellphone (C) and don't own cellphone (~C)

Since the question concerns percents only, let's say there are 100 employees in this population.
So, we get:


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

Let's say that x employees have laptops. This means that 100 - x employees don't have laptops. So we get:


The number of employees without laptops is 40% less than the employees with laptops.
In other words, the number of employees without laptops = 60% of the number of employees with laptops.
We can write: 100 - x = 0.6x
Simplify: 100 = 1.6x
x = 62.5
In other words, 62.5 employees have laptops.
This means that 37.5 employees don't have laptops. We get:


44% of the employees that own laptops do not own a cellphone
We now know that 62.5 employees have laptops, so 44% of 62.5 employees do not own a cellphone
44% of 62.5 = 27.5, so we get:


30% of the employees that own cellphone do not own laptops.
Let's let y = the total number of employees that own cellphone
We get:


If 30% of the employees that own a cellphone do not own laptops, then 70% of the employees that own a cellphone do own laptops.
So, 70% of y = the number of employees that own a cellphone and own a laptop.
Since we already know that 35 employees own a cellphone and own a laptop, so we can write 0.7y = 35
Solve to get y = 50
So, 50 employees own a cellphone, which means 50 employees don't own a cellphone. We get:



From here, we can fill in the remaining boxes:


Looks like 22.5 employees (out of 100 employees) own neither a laptop nor a cellphone.

Answer: B

Cheers,
Brent

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

Once you’re familiar with the Double Matrix Method, you can attempt these additional practice questions:

Easy Problem Solving questions
- http://www.beatthegmat.com/the-aam-aadmi-party-t272242.html
- http://www.beatthegmat.com/finance-majors-non-finance-majors-overlapping-set-question-t167425.html

Medium Problem Solving questions
- http://www.beatthegmat.com/probability-question-t273360.html
- http://www.beatthegmat.com/posted-speed-limit-t272374.html
- http://www.beatthegmat.com/motel-t271938.html
- http://www.beatthegmat.com/of-the-applicants-passes-a-certain-test-15-t270255.html
- http://www.beatthegmat.com/opening-night-and-late-patrons-t264869.html
- http://www.beatthegmat.com/ds-french-japanese-t222297.html

Difficult Problem Solving questions
- http://www.beatthegmat.com/ratio-problem-t268339.html
- http://www.beatthegmat.com/overlapping-sets-question-t265223.html
- http://www.beatthegmat.com/fractions-t264254.html
- http://www.beatthegmat.com/overlapping-sets-t264092.html
- http://www.beatthegmat.com/mba/2011/05/09/random-double-matrix-question-2

Easy Data Sufficiency questions
- http://www.beatthegmat.com/for-what-percent-of-those-tested-for-a-certain-t270596.html
- http://www.beatthegmat.com/ds-quest-t187706.html

Medium Data Sufficiency questions
- http://www.beatthegmat.com/sets-matrix-ds-t271914.html
- http://www.beatthegmat.com/each-of-people-voted-once-in-an-election-x-t271375.html
- http://www.beatthegmat.com/a-manufacturer-t270331.html
- http://www.beatthegmat.com/in-costume-for-halloween-t269355.html
- http://www.beatthegmat.com/mba/2011/05/05/random-double-matrix-question-1

Difficult Data Sufficiency questions
- http://www.beatthegmat.com/double-set-matrix-question-t271423.html
- http://www.beatthegmat.com/sets-t269449.html
- http://www.beatthegmat.com/mba/2011/05/16/random-double-matrix-question-3

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Last edited by Brent@GMATPrepNow on Tue Mar 25, 2014 11:45 am; edited 1 time in total

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Atekihcan Master | Next Rank: 500 Posts
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Post Thu Jun 13, 2013 12:40 am
For two overlapping sets, total number of elements = elements only in 1st set + elements only in 2nd set + elements in both + elements in none

Assume that 100 employees own laptops.
So, 100 - 40 = 60 employees do not own laptops.
So, total number of employees = (100 + 60) = 160

Now, 44 employees own laptop but do not own cellphones.
So, 44 employees own only laptops.
So, (100 - 44) = 56 employees own both.

Now, if the number of employees who own cellphones is N, 70% of N own both laptops and cellphone.
So, 70% of N = 56
So, 30% of N = (56/7)*3 = 24
So, 24 employees own only cellphone.

So, total = only laptop + only cellphone + both + none
So, 160 = 44 + 24 + 56 + none
So, none = 160 - 124 = 36

So, required percentage = (36/160)*100 = 22.5

Answer : B

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GMAT/MBA Expert

Post Thu Jun 13, 2013 7:12 am
jose.mario.amaya wrote:
At company X, 44% of the employees that own laptops do not own cellphone, and 30% of the employees that own cellphone do not own laptops. If the number of employees without laptops is 40% less than the employees with laptops, then what percent of the company's employees neither own a laptop nor a cellphone ??

A. 17.5
B. 22.5
C. 24
D. 26
E. 28

B
This is an EITHER/OR problem.
Every employee EITHER owns a laptop OR doesn't.
Every employee EITHER owns a cell or OR doesn't.
To organize the data, draw a GROUP GRID:

In the grid above, L = laptop, NL = no laptop, C = cell, NC = no cell.

44% of the employees who own a laptop do not own a cellphone.
Thus, 56% of laptop owners have BOTH:
.56L = LC.

30% of the employees who own a cellphone do not own a laptop.
Thus, 70% of cell owners have BOTH:
.7C = LC.

Since .56L = LC and .7C = LC, we get:
.56L = .7C
56L = 70C
4L = 5C
L/C = 5/4.

Let L = 50 and C = 40.
Since 70% of cell owners have both a laptop and a cell, LC = .7(40) = 28.
Enter these values into the grid and calculate any other values that can be determined:


The number of employees without laptops is 40% less than the employees with laptops.
Since L = 50, NL = 50 - .4(50) = 30.
Complete the grid:


Of the 80 total employees, 18 have neither a laptop nor a cell:
18/80 * 100 = 22.5%.

The correct answer is B.

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gmatdriller Master | Next Rank: 500 Posts Default Avatar
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Post Mon Mar 24, 2014 10:39 pm
GmatGuru, your Explanations are greatly appreciated as always.

Meanwhile, from L/C = 5/4, what determines the correct values of
L and C? I put this question because different values of L and C yield
different results.

If L/C = 25 / 20 (maintaining same ratio), 0.7 of 20 gives 14

Your explanation is appreciated.

Thanks.

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