double matrix

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[email protected]: Master | Next Rank: 500 Posts; Posts: 117; Joined: Mon Oct 27, 2008 5:08 pm; Thanked: 1 times

by [email protected] » Fri Dec 19, 2008 3:46 pm

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?

15

Can someone explain why the double matrix is not working for this problem?

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Source: Beat The GMAT — Problem Solving | Fri Dec 19, 2008 3:46 pm

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by Brent@GMATPrepNow » Fri Dec 19, 2008 5:28 pm

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?

The double matrix should work. We are told that 140 households do not use Brand B soap (60 of them use Brand A and 80 use neither).
This means that 60 households use Brand B (some use Brand A as well and some do not use Brand A)

Of the 60 Brand B users, we're told that the ratio of Brand A users to people who don't use Brand A is 1:3

When we divide 60 into a 1:3 ratio we get 15:45
So, 15 households use both brands.

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amitabhprasad: Master | Next Rank: 500 Posts; Posts: 259; Joined: Thu Jan 18, 2007 8:30 pm; Thanked: 16 times

by amitabhprasad » Sat Dec 20, 2008 12:21 am

Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15

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Bidisha800: Master | Next Rank: 500 Posts; Posts: 344; Joined: Sun Sep 28, 2008 11:00 amFollowed by: 1 member; Location: USA; Thanked: 6 times

by Bidisha800 » Fri Jan 02, 2009 12:01 am

amitabhprasad wrote:Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15

According to the set theory:

Total = (a) + (b) -both+ none

200 =60 + 3x -x +80
x= 30

where I went wrong ?

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DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Fri Jan 02, 2009 12:22 am

[quote="Bidisha800"][quote="amitabhprasad"]Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15[/quote]

According to the set theory:

Total = (a) + (b) -both+ none

200 =60 + 3x -x +80
x= 30

where I went wrong ?[/quote]

It's not Total = (a) + (b) - both + none.
The correct formula is Total = (a) + (b) + both + none and then you get the right answer of x = 15. Notice that you do not have "at least" anywhere in the problem, therefore your formula is not correct.

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ronniecoleman: Legendary Member; Posts: 546; Joined: Sun Nov 16, 2008 11:00 pm; Location: New Delhi , India; Thanked: 13 times

by ronniecoleman » Fri Jan 02, 2009 4:23 am

Bidisha800 wrote:
amitabhprasad wrote:Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15
According to the set theory:

Total = (a) + (b) -both+ none

200 =60 + 3x -x +80
x= 30

where I went wrong ?

very common mistake we make

a U b = a + b - a <intersection> b

but here a is (total a) that is (only a + both a and b )

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pandeyvineet24: Legendary Member; Posts: 527; Joined: Mon Jun 02, 2008 9:14 am; Location: Atlanta; Thanked: 17 times

by pandeyvineet24 » Fri Jan 02, 2009 6:37 am

Double Matrix soln

A Not A Total

B x 3x 60

Not B 60 80 140

total x+60 3x+80 200

solve for x.

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DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Fri Jan 16, 2009 1:02 am

Explanation for how I solved this one:
Notice that any given household has 4 options:
- they use both brands of soap - x
- they use only brand A - 60 hh
- they use only brand B - 3x
- they don't use any of the two - 80 hh
Since you do not see "at least" anywhere in the problem, then adding the number of household that picked either one of the four would give you the sum of 200 households that participated in the survey. So you will have:

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

Be careful though: if "at least" appears anywhere in the problem, that changes thing quite a bit. Because if a hh says it uses at least brand B of soap, it could use only brand B, but it could also use both brands.

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sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Tue Feb 10, 2009 10:18 pm

Folks, you can avoid confusion if you can use a simple venn diagram.

Have a look at this....

Once you master Venn diagrams I think you can avoid drawing them at least in the case where there are two variables.

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hutch27: Master | Next Rank: 500 Posts; Posts: 141; Joined: Fri Sep 07, 2012 10:25 pmFollowed by: 1 member; Location: NY; Thanked: 5 times

by hutch27 » Tue Mar 26, 2013 11:28 am

How do you know the ratio is 1 to 3 though? The problem says that "for every household that used both brands of soap , 3 used only brand B"

So doesn't this mean that out of all the households that used both A and B, 3 only used brand B?

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Mar 26, 2013 11:39 am

hutch27 wrote:How do you know the ratio is 1 to 3 though? The problem says that "for every household that used both brands of soap , 3 used only brand B"

So doesn't this mean that out of all the households that used both A and B, 3 only used brand B?

The phrase in red is incorrect.
For every X, there are 3 Y's.
This means that for every ONE element that is an X, THREE elements are Y's.
In other words:
X:Y = 1:3.

In the problem above:
For every household that used both brands of soap, 3 used only brand B.
In math terms:
(both soaps) : (only B) = 1:3.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Mar 27, 2013 5:15 am

Incidentally, I thought I'd point out (to those unfamiliar with the Double Matrix Method) that this technique can be used for most questions featuring a population in which each member has two criteria associated with it.
Here, the criteria are:
- using or not using Brand A soap
- using or not using Brand A soap

For more information about the Double Matrix Method and some additional practice questions, check out these 3 BTG articles:

- 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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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