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

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tar32 Newbie | Next Rank: 10 Posts Default Avatar
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Venn Diagram vs. Formula [Grp 1 + Grp 2 - Both + Neither ]

Post 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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    anshumishra Legendary Member
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    Post 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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    Post 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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    Post 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?

    anshumishra Legendary Member
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    Post 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

    Post 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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    GMAT Private Tutor
    GMATGuruNY@gmail.com
    If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
    Available for tutoring in NYC and long-distance.
    For more information, please email me at GMATGuruNY@gmail.com.

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    showbiz Senior | Next Rank: 100 Posts
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    Post 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.

    http://www.manhattangmat.com/forums/post4973.html

    ankur.agrawal Master | Next Rank: 500 Posts
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    Post 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.

    KristenH88 Junior | Next Rank: 30 Posts Default Avatar
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    Post 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.

    KristenH88 Junior | Next Rank: 30 Posts Default Avatar
    Joined
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    Post Fri Jan 10, 2014 4:11 pm
    Nevermind, "showbiz" below gave the PERFECT link, those with my question should read it asap. Smile 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.

    Post 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"):


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


    60 used only Brand A soap
    We get...


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


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


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


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


    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: http://www.gmatprepnow.com/module/gmat-word-problems?id=919

    Then try these additional practice questions that can be solved using the Double Matrix Method:
    - http://www.beatthegmat.com/mba/2011/05/05/random-double-matrix-question-1
    - http://www.beatthegmat.com/mba/2011/05/09/random-double-matrix-question-2
    - http://www.beatthegmat.com/mba/2011/05/16/random-double-matrix-question-3
    - http://www.beatthegmat.com/ds-quest-t187706.html
    - http://www.beatthegmat.com/overlapping-sets-questions-t183320.html
    - http://www.beatthegmat.com/finance-majors-non-finance-majors-overlapping-set-question-t167425.html
    - http://www.beatthegmat.com/ds-french-japanese-t222297.html
    - http://www.beatthegmat.com/sets-t269449.html#692540
    - http://www.beatthegmat.com/in-costume-for-halloween-t269355.html#692116

    Cheers,
    Brent

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    smkhan Junior | Next Rank: 30 Posts Default Avatar
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    Post 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.

    Post 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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    smkhan Junior | Next Rank: 30 Posts Default Avatar
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    Post 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

    Post 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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