Problem Solving

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

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

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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 this video: https://www.gmatprepnow.com/module/gmat- ... ems?id=919

Once you're familiar with this technique, you can attempt these additional practice questions:

Easy Problem Solving questions
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Medium Problem Solving questions
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Difficult Problem Solving questions
- https://www.beatthegmat.com/ratio-problem-t268339.html
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Easy Data Sufficiency questions
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- https://www.beatthegmat.com/ds-quest-t187706.html

Medium Data Sufficiency questions
- https://www.beatthegmat.com/sets-matrix-ds-t271914.html
- https://www.beatthegmat.com/each-of-peop ... 71375.html
- https://www.beatthegmat.com/a-manufacturer-t270331.html
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- https://www.beatthegmat.com/mba/2011/05/ ... question-1

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

Msaad 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

This is an overlapping set question. A great way to solve this problem would be to set up a table with two categories: soap A and soap B. More specifically, our table will be labeled as follows:

1) Used Soap A

2) Did not use Soap A

3) Used Soap B

4) Did not use soap B

We are given:

Total = 200

Did not use soap A nor soap B = 80

Used only soap A (meaning used A but not B) = 60

We are also given that for every household that used both brands of soap, 3 used only brand B soap. Thus, we can say the following:

Used both soaps = x

Used only soap B (meaning B but not A) = 3x

Let's fill all of this into a table. Remember, all the columns and rows sum together.



From our table above, we can create the following equation:

4x + 140 = 200

4x = 60

x = 15

Thus, 15 people use both soap A and soap B.

Answer: A