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Msaad: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Fri Nov 11, 2016 6:28 am

venn Diagram Question

by Msaad » Fri Nov 11, 2016 6:29 am

Guys,

Can you please help me out with this question.

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?

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Source: Beat The GMAT — Problem Solving | Fri Nov 11, 2016 6:29 am

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by GMATGuruNY » Fri Nov 11, 2016 6:35 am

A couple of approaches are offered here:
https://www.beatthegmat.com/calculating- ... 83812.html

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DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 am; Location: Boston, MA; Thanked: 1153 times; Followed by:128 members; GMAT Score:770

by DavidG@VeritasPrep » Fri Nov 11, 2016 6:48 am

Msaad wrote:Guys,

Can you please help me out with this question.

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

In addition to the approaches in the link Mitch posted, you could back-solve
.

Try B. If 20 people used both kinds of soap, and for every one that used both, 3 used only soap B, then 60 used only B.
So now we'd have the following:

Used neither: 80
Used only A: 60
Used only B: 60
Used Both: 20

Adding, we'd get 220. But we know there are actually 200 households surveyed. So B, 20, is too large. If the answer must be smaller, it's A.

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DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 am; Location: Boston, MA; Thanked: 1153 times; Followed by:128 members; GMAT Score:770

by DavidG@VeritasPrep » Fri Nov 11, 2016 6:50 am

(And though you don't have to do additional work, if you wanted to validate that 15 works...)

If 15 people used both kinds of soap, and for every one that used both, 3 used only soap B, then 45 used only B.
So now we'd have the following:

Used neither: 80
Used only A: 60
Used only B: 45
Used Both: 15

Adding, we'd get 200, so we know A is correct.

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

by Brent@GMATPrepNow » Fri Nov 11, 2016 8:55 am

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

Brent Hanneson - Creator of GMATPrepNow.com

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by Matt@VeritasPrep » Fri Nov 11, 2016 2:06 pm

You could also say that

Neither + Only A + Only B + Both = 200

or

80 + 60 + 3x + x = 200

That saves a lot of time drawing circles and boxes!

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Quote

Msaad: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Fri Nov 11, 2016 6:28 am

by Msaad » Sat Nov 12, 2016 1:05 am

Thanks a lot guys. But i couldnt understand that why are we putting 3x. I mean in the question there is nothing that says that it is 3 times or something like it. It just says that it is 3 only used brand B.

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

by GMATGuruNY » Sat Nov 12, 2016 4:44 am

Msaad wrote:Thanks a lot guys. But i couldnt understand that why are we putting 3x. I mean in the question there is nothing that says that it is 3 times or something like it. It just says that it is 3 only used brand B.

For every dollar that John saved, he spent five dollars.
This wording expresses a RATIO:
If John saved $1, then he spent $5.
If John saved $2, then he spent $10.
If John saved $3, then he spent $15.
In general:
If John saved x dollars, then he spent 5x dollars.

In the problem above:
For every household that used both brands of soap, 3 used only Brand B soap.
Thus:
If x households used both brands, then 3x households used only B.

Quote

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by Jeff@TargetTestPrep » Sat Nov 12, 2016 5:50 am

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.