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

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tar32 Just gettin' started! 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 GMAT Destroyer!
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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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    GMATGuruNY@gmail.com
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    diebeatsthegmat GMAT Titan Default Avatar
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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 GMAT Destroyer!
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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

    GMAT/MBA Expert

    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.

    _________________
    Mitch Hunt
    GMAT Private Tutor
    GMATGuruNY@gmail.com
    If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
    Contact me about long distance tutoring!

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    showbiz Rising GMAT Star
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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 Really wants to Beat The GMAT!
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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 Just gettin' started! 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 Just gettin' started! Default Avatar
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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 Just gettin' started! 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 Just gettin' started! 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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    Study Smart! Use Beat The GMAT’s FREE 60-Day Study Guide in conjunction with GMAT Prep Now’s video course and reach your target score in 2 months! With two money-back guarantees, you can try us out risk-free.

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