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Venn Diagram vs. Formula [Grp 1 + Grp 2 - Both + Neither ]

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by tar32 » Sat Dec 18, 2010 2:40 pm

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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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
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by anshumishra » Sat Dec 18, 2010 4:11 pm
tar32 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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
Both are good, as long as you are comfortable with them .
I have used Venn diagram to solve this question :

So, 200 = 60 + x+ 3x + 80 => x = 15

Without Venn diagram :

Total = Group 1 + Group2 - Both + Neither
=> 200 = (60+x) + (3x+x) - x + 80 = 60 + 4x + 80 => x = 15.

Thanks
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by GMATGuruNY » Sat Dec 18, 2010 4:50 pm
tar32 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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
The formula above is not the best approach for this problem. In the formula above, Group 1 = (households that used only Brand A + households that used both Brand A and Brand B). The problem gives us the number of households that used only Brand A (60). There is no easy way to plug this value into the formula.

Here is a formula that would work for this problem:

Total = Only Brand A + Only Brand B + Both + Neither

Total = 200
Only Brand A = 60
Neither = 80
Both = x
Only Brand B = 3x

Plugging these values into the formula, we get:

200 = 60 + 3x + x + 80
60 = 4x
x = 15.

The correct answer is A.
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by diebeatsthegmat » Sun Dec 19, 2010 6:30 pm
GMATGuruNY wrote:
tar32 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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
The formula above is not the best approach for this problem. In the formula above, Group 1 = (households that used only Brand A + households that used both Brand A and Brand B). The problem gives us the number of households that used only Brand A (60). There is no easy way to plug this value into the formula.

Here is a formula that would work for this problem:

Total = Only Brand A + Only Brand B + Both + Neither

Total = 200
Only Brand A = 60
Neither = 80
Both = x
Only Brand B = 3x

Plugging these values into the formula, we get:

200 = 60 + 3x + x + 80
60 = 4x
x = 15.

The correct answer is A.
hi, can you please explain me why the brand B is 3x??? it says that only 3 household in both used brand B. does it mean that B=3?
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by anshumishra » Sun Dec 19, 2010 6:34 pm
diebeatsthegmat wrote:
GMATGuruNY wrote:
tar32 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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
The formula above is not the best approach for this problem. In the formula above, Group 1 = (households that used only Brand A + households that used both Brand A and Brand B). The problem gives us the number of households that used only Brand A (60). There is no easy way to plug this value into the formula.

Here is a formula that would work for this problem:

Total = Only Brand A + Only Brand B + Both + Neither

Total = 200
Only Brand A = 60
Neither = 80
Both = x
Only Brand B = 3x

Plugging these values into the formula, we get:

200 = 60 + 3x + x + 80
60 = 4x
x = 15.

The correct answer is A.
hi, can you please explain me why the brand B is 3x??? it says that only 3 household in both used brand B. does it mean that B=3?
That is because the question says :
For every household that used both brands of soap, 3 used only Brand B soap

That means if x households used both the brands, then 3x used brand B.

Thanks
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by GMATGuruNY » Sun Dec 19, 2010 9:01 pm
diebeatsthegmat wrote:
GMATGuruNY wrote:
tar32 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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
The formula above is not the best approach for this problem. In the formula above, Group 1 = (households that used only Brand A + households that used both Brand A and Brand B). The problem gives us the number of households that used only Brand A (60). There is no easy way to plug this value into the formula.

Here is a formula that would work for this problem:

Total = Only Brand A + Only Brand B + Both + Neither

Total = 200
Only Brand A = 60
Neither = 80
Both = x
Only Brand B = 3x

Plugging these values into the formula, we get:

200 = 60 + 3x + x + 80
60 = 4x
x = 15.

The correct answer is A.
hi, can you please explain me why the brand B is 3x??? it says that only 3 household in both used brand B. does it mean that B=3?
As mentioned above, for every household that used both brands, 3 used only Brand B means that the ratio of both:only B = 1:3. Thus, if x used both brands, 3x used only Brand B.
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As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
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by showbiz » Mon Dec 20, 2010 7:21 pm
The key to this question are the words "Only Brand A"

The Venn diagram drawn above puts 60 and x in one circle, which doesn't apply in this case. In essence, you would have to take the middle slice (x) separately from 60 and 3x.

https://www.manhattangmat.com/forums/post4973.html
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by ankur.agrawal » Mon Jan 17, 2011 1:04 am
I am facing a hard time analyzing Questions based on SET theory, Venn Diagrams.

Sumbody pls suggest the way out. Concepts, Practice question anythg that can help.

Thanx in advance.
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by KristenH88 » Fri Jan 10, 2014 4:01 pm
Why is neither not subtracted in this case, and when do you know to use the original a+b-both+neither=total instead of adding them all? In this Diag I got it wrong using the formula I learned and tried yours and got it right. When to use? Thank you.
GMATGuruNY wrote:
tar32 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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
The formula above is not the best approach for this problem. In the formula above, Group 1 = (households that used only Brand A + households that used both Brand A and Brand B). The problem gives us the number of households that used only Brand A (60). There is no easy way to plug this value into the formula.

Here is a formula that would work for this problem:

Total = Only Brand A + Only Brand B + Both + Neither

Total = 200
Only Brand A = 60
Neither = 80
Both = x
Only Brand B = 3x

Plugging these values into the formula, we get:

200 = 60 + 3x + x + 80
60 = 4x
x = 15.

The correct answer is A.
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by KristenH88 » Fri Jan 10, 2014 4:11 pm
Nevermind, "showbiz" below gave the PERFECT link, those with my question should read it asap. :) Thank you showbiz.
GMATGuruNY wrote:
tar32 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


*which is the better approach? Venn vs. Formula [Group 1 + Group 2 - Both + Neither = Total ]
The formula above is not the best approach for this problem. In the formula above, Group 1 = (households that used only Brand A + households that used both Brand A and Brand B). The problem gives us the number of households that used only Brand A (60). There is no easy way to plug this value into the formula.

Here is a formula that would work for this problem:

Total = Only Brand A + Only Brand B + Both + Neither

Total = 200
Only Brand A = 60
Neither = 80
Both = x
Only Brand B = 3x

Plugging these values into the formula, we get:

200 = 60 + 3x + x + 80
60 = 4x
x = 15.

The correct answer is A.
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Venn Diagram vs. Formula [Grp 1 + Grp 2 - Both + Neither ]

by Brent@GMATPrepNow » Sat Jan 11, 2014 6:49 am
tar32 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
We can also 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 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

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

Then try these 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
Last edited by Brent@GMATPrepNow on Fri Apr 13, 2018 11:38 am, edited 1 time in total.
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by smkhan » Mon Sep 08, 2014 7:32 am
Hi,

Why is the group formula (Group 1 + Group 2 - Both + Neither = Total )not suitable for this problem as pointed out by Mitch. It reduces the equation to the same equation as you would get if you use Venn diagram. 60+4x+80=200. Just trying to clarify the concept.
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by Brent@GMATPrepNow » Mon Sep 08, 2014 7:40 am
smkhan wrote:Hi,

Why is the group formula (Group 1 + Group 2 - Both + Neither = Total )not suitable for this problem as pointed out by Mitch. It reduces the equation to the same equation as you would get if you use Venn diagram. 60+4x+80=200. Just trying to clarify the concept.
It might help us if you explain how 60+4x+80=200 is related to Group 1 + Group 2 - Both + Neither = Total
In your equation there are only 3 terms (60+4x+80) on the left side, yet there are 4 terms (Group 1 + Group 2 - Both + Neither) on the left side of the group formula.

The transition from 4 terms to 3 terms is what makes it tricky to apply the formula here.

Cheers,
Brent
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by smkhan » Mon Sep 08, 2014 2:44 pm
Hi,

Sorry should have written the solution but it was the same as anshumishra's at the top that's why I didnt write it. But here's how I solved it first using the group formula and than with Venn diagram.

A' alone - 60
A&B both - x
B' alone - 3x
N - Neither A nor B - 80
A - Total Brand A, 60+x
B - Total Brand B, 3x+x

Using the group formula, A+B-A&B+N=200

(60+x)+(3x+x)-x+80=200
60+4x+80=200
4x=200-140=60
x=15

Using Venn diagarm, Total = Only Brand A + Only Brand B + Both + Neither
200= 60+3x+x+80
200=60+4x+80

Thanks
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by Brent@GMATPrepNow » Mon Sep 08, 2014 2:48 pm
That's perfect, smkhan.

However, as you can see, there's quite a bit of extra reasoning beyond just plugging numbers into the formula. That's why Mitch suggested that the "Group 1 + Group 2 - Both + Neither = Total" formula might not be the easiest route.

Cheers,
Brent
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