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Diagnostic Test Q6 - Overlapping Sets

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Diagnostic Test Q6 - Overlapping Sets

by Bullzi » Fri Sep 18, 2015 2:52 am
Hello,

So, I have an issue with arriving at an answer the OG gives for a Diagnostic Test question using a method I am comfortable with. While I could understand OG's answer steps, I couldn't make out why my approach wouldn't work. It would be helpful if you could help with pointers

The question is No. 6 on overlapping sets,

A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?

(A) 15
(B) 20
(C) 30
(D) 40
(E) 45

And, below is the approach I used,

1. I was comfortable using a table structure rather than a Venn diagram, so, I draw a 3X3 table
2. Based on the data in the question, I fill numbers and for the unknowns, I assign and use a variable (as in the image)
3. Based on the table, i arrive at an equation thus, 60-x + 80 = 200 - 3x solving which I arrive at 30 as the value for x. However, the correct result as per OG is 15

What am I missing..?! I appreciate all help!

Thanks,
Bullzi
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Diagnostic Q6 - 3X3 table
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by Brent@GMATPrepNow » Fri Sep 18, 2015 7:41 am
A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?

A: 15
B: 20
C: 30
D: 40
E: 45
Here, we have a population of 200 households , and the two characteristics are:
- using or not using Brand A soap
- using or not using Brand B soap

So, we can set up our matrix as follows (where "~" represents "not"):
Image

80 used neither Brand A nor Brand B soap
We can add this to our diagram as follows:
Image

60 used only Brand A soap
We get...
Image

At this point, we can see that the right-hand column adds to 140, which means 140 households do NOT use brand B soap.
Image

Since there are 200 households altogether, we can conclude that 60 households DO use brand B soap.
Image

For every household that used BOTH brands of soap...
Let's let x = # of households that use BOTH brands....
Image

...3 used only Brand B soap.
So, 3x = # of households that use ONLY brand B soap
Image

At this point, when we examine the left-hand column, we can see that x + 3x = 60
Simplify to get 4x = 60
Solve to get x = 15

How many of the 200 households surveyed used BOTH brands of soap?
Since x = # of households that use BOTH brands of soap, the correct answer here is A

------------------------------------
Overlapping sets questions are 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:

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- https://www.beatthegmat.com/the-aam-aadm ... 72242.html
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Difficult Problem Solving questions
- https://www.beatthegmat.com/ratio-problem-t268339.html
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Easy Data Sufficiency questions
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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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by Brent@GMATPrepNow » Fri Sep 18, 2015 7:44 am
Bullzi wrote:Hello,

So, I have an issue with arriving at an answer the OG gives for a Diagnostic Test question using a method I am comfortable with. While I could understand OG's answer steps, I couldn't make out why my approach wouldn't work. It would be helpful if you could help with pointers

The question is No. 6 on overlapping sets,

A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?

(A) 15
(B) 20
(C) 30
(D) 40
(E) 45

And, below is the approach I used,

1. I was comfortable using a table structure rather than a Venn diagram, so, I draw a 3X3 table
2. Based on the data in the question, I fill numbers and for the unknowns, I assign and use a variable (as in the image)
3. Based on the table, i arrive at an equation thus, 60-x + 80 = 200 - 3x solving which I arrive at 30 as the value for x. However, the correct result as per OG is 15

What am I missing..?! I appreciate all help!

Thanks,
Bullzi
The question says that 60 used only Brand A soap
This means 60 used A but did NOT use B.

You placed 60-x in the box that should have a 60 in it.

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Bullzi » Fri Sep 18, 2015 9:24 am
Thanks Brent for your response

I went back to my notes and material to double-check the strategy

Looks like I may have interpreted the question incorrectly. I interpreted the text as saying that the total number of households using Soap A (including the ones that might use Soap B) was 60. Translating this understanding into an equation, I used x for the intersection (households using either of the brands) and 60-x for the households that used only A

Now, I am just curious, had the problem framed the text thus 'of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used (without using the word ONLY from the original text) Brand A soap', I am assuming my original approach would've worked just alright as this count would've included both the intersection of A and B and A's own individual count too. Is my understanding correct?

Thanks!
Bullzi
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by gmatbeater1989 » Tue Oct 20, 2015 2:44 pm
Bullzi wrote:

A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?

(A) 15
(B) 20
(C) 30
(D) 40
(E) 45


What's wrong with my answer?

Total = group A + group B - both + neither
200 = 60 + 3 - x + 80
200 = 143 - x
x = -57
I know it's wrong, but I don't see why.
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by theCEO » Tue Oct 20, 2015 3:10 pm
gmatbeater1989 wrote:
Bullzi wrote:

A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?

(A) 15
(B) 20
(C) 30
(D) 40
(E) 45


What's wrong with my answer?

Total = group A + group B - both + neither
200 = 60 + 3 - x + 80
200 = 143 - x
x = -57
I know it's wrong, but I don't see why.


Total = group A + group B - both + neither
200 = (group A only + both group a and b) + (group b only + both group a and b) - both group a and b + neither
200 = (60 + x) + (3x + x) - x + 80
200 = 140 + 4x
x = 15
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by gmatbeater1989 » Tue Oct 20, 2015 3:45 pm
theCEO wrote:
gmatbeater1989 wrote:
Bullzi wrote:

A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?

(A) 15
(B) 20
(C) 30
(D) 40
(E) 45


What's wrong with my answer?

Total = group A + group B - both + neither
200 = 60 + 3 - x + 80
200 = 143 - x
x = -57
I know it's wrong, but I don't see why.


Total = group A + group B - both + neither
200 = (group A only + both group a and b) + (group b only + both group a and b) - both group a and b + neither
200 = (60 + x) + (3x + x) - x + 80
200 = 140 + 4x
x = 15

Aha!
I didn't realize I also missed the part that says "for every household that used both brands of soap,"
THANKS!
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