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
The OA is A.
Source: Economist GMAT
A marketing firm determined that, of 200 households surveyed
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Given that ;
Those who used neither brand A nor B =80
Those who used brand A only = 60
Let those who used both brands of soap = x
For every household that uses both brands,
3 households used only brand B soap
I.e if 1 household uses both soap 3 other household uses brand B soap = 3x
Total house hold surveyed = 200
Therefore, 80 + 80 + x + 3x = 200
140 + 4x = 200
4x = 200 - 140
4x = 60
Divide both sides by co-efficient of x,
$$\frac{4x}{4}=\ \frac{60}{4}$$
Those who used neither brand A nor B =80
Those who used brand A only = 60
Let those who used both brands of soap = x
For every household that uses both brands,
3 households used only brand B soap
I.e if 1 household uses both soap 3 other household uses brand B soap = 3x
Total house hold surveyed = 200
Therefore, 80 + 80 + x + 3x = 200
140 + 4x = 200
4x = 200 - 140
4x = 60
Divide both sides by co-efficient of x,
$$\frac{4x}{4}=\ \frac{60}{4}$$
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We can also solve this question using the Double Matrix Method.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
------------------------------------
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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Cheers,
Brent
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Excellent opportunity to use Venn diagrams (a.k.a. "overlapping sets")!swerve 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
Source: Economist GMAT
$$? = x$$
$$120 = 60 + x + 3x\,\,\,\,\, \Rightarrow \,\,\,\,? = x = 15$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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This is an overlapping set question. We can use the following formula:swerve 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
Total = A only + B only + Both + Neither
We are given that the Total = 200, A only = 60, and Neither = 80. We are given that for every household that used both brands of soap, 3 used only Brand B. So, if we let x = Both, then 3x = B only. Thus:
200 = 60 + 3x + x + 80
200 = 140 + 4x
60 = 4x
x = 15
Thus, 15 households used both brands of soap.
Answer: A
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