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Calculating Mixed Groups

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Calculating Mixed Groups

by infiniti007 » Mon Jun 08, 2015 7:16 pm
Hi, I have a question on the following problem. In particular, I'm wondering why I cannot use the Mixed Groups Formula to solve this problem. Or, if I can, how have I applied it incorrectly?

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

Total = Group 1 + Group 2 - Both + Neither

200 = 60 + 3x - x + 80
120 = 60 + 2x
2x = 60
x = 30

Where did I go wrong?
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by Brent@GMATPrepNow » Mon Jun 08, 2015 7:28 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?

A: 15
B: 20
C: 30
D: 40
E: 45
We can also solve this question using the Double Matrix Method.

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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Difficult Problem Solving questions
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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
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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
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by Brent@GMATPrepNow » Mon Jun 08, 2015 7:31 pm
infiniti007 wrote:Hi, I have a question on the following problem. In particular, I'm wondering why I cannot use the Mixed Groups Formula to solve this problem. Or, if I can, how have I applied it incorrectly?

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

Total = Group 1 + Group 2 - Both + Neither

200 = 60 + 3x - x + 80
120 = 60 + 2x
2x = 60
x = 30

Where did I go wrong?
We're told that 60 used only Brand A soap.
The formula requires us to know the number that used Brand A soap, which included people who use ONLY brand A AND people who use both brands A and B.

The same applies to Brand B soap.

Cheers,
Brent
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by GMATGuruNY » Mon Jun 08, 2015 8:01 pm
infiniti007 wrote:Hi, I have a question on the following problem. In particular, I'm wondering why I cannot use the Mixed Groups Formula to solve this problem. Or, if I can, how have I applied it incorrectly?

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

Total = Group 1 + Group 2 - Both + Neither
As Brent has mentioned, 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).
But the prompt 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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