A marketing firm determined that, of 200 households surveyed, 80 used

60(number of brand A) + X(number of households that use both brans)+ 3x(number of households that use only brand B) + 80 = 200

X = 15

Shouldn't the answer be [C]

60(Brand A) +3x(Brand B)-x(Both Brands) = 200-80(total is 120 since 80 use neither)

60+2x = 120 => x =30

No pallavisatsangi
The answer is A (15) you can not subtract x (both using A and B) Because in the question it is mentioned that 60 people is only brand A and accordingly 3x people use only Brand B.... if the term 'ONLY' had not been used in the question then you would have subtracted x .. but since in both cases term 'only' has been used, you can not subtract x
So eq. becomes
60 +x+3x+80=200
x=15

Nice question, i took around 1 minute to comprehend this line "for every household that used both brands of soap, 3 used only Brand B soap" and still over looked the word "only" Brand B. I chose (C)30 , but it should be 15. Nice explanation every one.

One important thing i observed in these questions, in which we have to deal with Only A/ Only B type issues that it is better to go with Venn-Dia. rather than table as it adds to complexity (only A-->not B)and consumes more time than former.
On the other hand, the table is easier to work with when the questions deal with "both/neither-nor" elements, range of elements. Just my opinion.

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

If someone could tell me how to solve the problem using the manhattan method (with grids), that would be awesome. For such a simple problem, I can't figure out how I'm messing up setting up the grid.

The data in your table should be as follow:

x 3x 60
60 80 140
60+x 80+3x 200

From the first line, x + 3x = 60 (the same equation can be obtained using the last line - 60 + x + 80 + 3x = 200).

Your mistake was misplacing 60 - it is the number of households using only brand A (meaning A but not B).

Hi

I have not been able to understand question 6 of the Quantitative section explanation at page 48.



Question 6 explanation: Arithmetic operations on rational numbers NOTE I copied exactly how GMAT written the question and explanation[/b]

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

[Their explanation]

Since it is given that 80 households use neither Brand A nor Brand B, then 200-80 = 120 (UNTIL HERE IT IS CORRECT) must use Brand A, Brand B, or both. It is also given that 60 households use only Brand A [b](CORRECT) and that three times as many households use both brands (WRONG? IN THE DATA GIVEN IT SAYS ONLY 3 PEOPLE USE BRAND B) [/b]. If x is the number of households that use both Brand A and Brand B, then 3x use Brand B alone. A Venn diagram can be helpful for visualizing the logic of the given information for this item:

Brand A Brand B

60 x 3x (I HAD ANSWER B. 20 60 is given 3 is given so 20 times 3 = 60 However, if we have 120 people who used Brand A (60) and B (3) than subtract 120 - 60 - 3 = 57 is left so they must use both Brand A & B. This is how I at first saw it but since non of these answers were possible it only makes sense to me the correct answer must be than 20)

All the sections in the circles can be added up and set equal to 120, and then the equation can be solved for x:

60+x+3x=120

60+4x=120 combine like terms (WHERE DOES THE 4 COMES FROM)

4x=60 subtract 60 from both sides

x=15 divided both sides by 4

Answer A


please explain to me if I am wrong. I will post more mistakes I have noticed in the book.

thanks
TD

Dear Don,

I'm happy to help with this.

I will say, I believe the OG is correct here and, I am sorry to say, you are mistaken. Keep in mind that this book, the GMAT OG, is probably proof-read dozen of times, and combed by expert after expert, and many of the problem have been recycled for many successive conditions now, so it's highly unlikely that it will be dripping with mistakes. If I may presume, I would suggest being careful leveling accusations of outright mistakes against it, only because it is already such well-trodden territory.

For this problem, consider the wording:
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?

I agree, the wording is not ideal, but at least it is unambiguous. We agree on the 80, the stinky houses that don't use soap --- let's focus on the last three groups

(1) households that use only Brand A soap = 60
(2) households that use both brands of soap
(3) households that use only Brand B soap

It's very important to realize that these three groups are disjoint, that is to say, non-overlapping and mutually exclusive. It would be absolutely impossible for any single household be to in any two of these groups simultaneously. Membership in any one automatically precludes the possibility of being either of the other two.

For the x houses in group (2), the houses that use both soaps, there are three times as many, 3x, that use only Brand B soap, and again, there is absolutely no possibility of overlap between this x and this 3x --- they have to be two completely different sets of household. Furthermore, the 60 households in (1) must have zero overlap with either the x or the 3x.

Therefore, when we add them, we get 60 + x + 3x = 60 + 4x.

Has this cleared up this question for you? Please let me know if you have any further questions.

Mike

Hi Mike,

Thanks for your response. However, even with your explanation I just don't agree and simply don't get it. Perhaps it is that I am new to GMAT and don't have a mathematical background but I believe in logic and the way they structured this question is not logical at all let alone their explanation. It feels like they are over analyzing the question with unseen data, expressing in a language that is not coherent in my opinion. I don't care if it is just me. But as a GMAT taker and I know I don't doubt my intelligence I think from what I have so far seen in the GMAT book is that their descriptions are not always explained in a clear fashion.

But that aside.

We both agree that:

A. total of 200 households surveyed
B. 80 used neither Brand A & B
C. 60 used only Brand A
D. - now it clearly says "and for every household that used both brands of soap, 3 used only Brand B soap.
E. used both Brand A and B ????


(3) households that use only Brand B soap (3 households used only Brand B it literally says this)



"For the x houses in group (2), the houses that use both soaps, there are three times as many, 3x, that use only Brand B soap, and again, there is absolutely no possibility of overlap between this x and this 3x --- they have to be two completely different sets of household. Furthermore, the 60 households in (1) must have zero overlap with either the x or the 3x. Therefore, when we add them, we get 60 + x + 3x = 60 + 4x"

Here I get lost with your explanation it just doesn't make sense to me. Where in the question does it say THREE TIMES AS MANY? It says 3 used only Brand B.

Thanks for trying to explain it to me but try again if you would like and perhaps in a more simplified way because it seems from GMAT explanation and yours that you guys see something in the wording that I don't see.

Ted

Ted

I think we have a basic semantic issue here.

If I say
1) Two students went to the hot dog stand, and three went to the pizza shack.

that is a very concrete statement about five students --- 3 in one group and 2 in another group. It's just about those five, nobody else. Mathematically, this is a finite statement --- there is only one set of numbers possible for the scenario it conveys.

BUT, if I say
2) "For every two students who go to the hot dog stand, three go to the pizza shack."

this is a very different category of statement. This is a statement about ratios and proportions. This statement means --- if I take all the students who go to the hot dog stand and group them by twos, then take all the students who go the pizza shack and group them by threes, there will be a 1-to-1 match of each hot dog pair and each pizza trio. From that statement alone, we have no idea what the total number of students involved is --- it could be in the dozens, or the hundreds, or the thousands. All this statement tells about are the proportions. The ratio of hot dog goer to pizza goers is 2:3. If we divide the entire group into five "parts", then two "parts" go to the hot dog shack, and three "parts" go to the pizza stand. The language "for every two students" emphatically does not mean that only two students were involved. Rather, it means we can break all the students who went to the hot dog stand into groups of two, pairs, and for each and every one of those pairs, there will be exactly three students who went to the pizza shack, and if we collect all those groups of three, it will add up magically to every single student who went to the pizza shack. This would work if
4 went to the hot dog stand, and 6 went to the pizza shack, or
20 went to the hot dog stand, and 30 went to the pizza shack, or
100 went to the hot dog stand, and 150 went to the pizza shack, or
146 went to the hot dog stand, and 219 went to the pizza shack, or
600 went to the hot dog stand, and 900 went to the pizza shack, etc etc etc (there are, of course, an infinite number of possibilities contained in that statement)
Fundamentally, this second statement is a statement that admits of infinite possibilities -- that is what puts it in an entirely different category from the first statement.

This is a special mathematical idiom you need to learn. When a question says

For every A people who do X, B people do Y.

That is inherently information not about absolute quantities but about ratios & proportions. The ratio of (people who did X) to (people who did Y) has to be A:B. Those first two words, "for every" --- that's the tip-off that this special mathematical idiom is being used.

Thus in this problem, when they say: "for every household that used both brands of soap, 3 used only Brand B soap" ----- this also is, inherently, a statement about ratios and proportions, and emphatically NOT a statement about, say, only 3 households. Rather it is saying, the ratio of (household that use both brands of soap) to (households that household that used both brands of soap) must be 1:3 --- in other words, there are three times as many of the latter as their are of the former. Again, from this statement in isolation, we don't have any idea of the absolute numbers involved ---- 5 & 15, or 10 & 30, or 100 & 300 --- we have to use the rest of the information in the problem to determine the actual values.

Does this make sense?

Also, my friend, I would caution you. From this misunderstanding, it would appear that you have several mathematics fundamentals to learn. That's fine ---- GMAT Club is an excellent place to ask questions, and the other "GMAT experts" and I are happy to help. But, I would caution you not to advertise differences between your understanding vs. the OG as "mistakes in the OG", if you see what I mean. Simply to accelerate your own learning, I would suggest a more question-oriented, everything-to-learn approach would most benefit your overall progress as a student.

Please let me know if you have any further questions about any of this.

Mike

Hi Mike

Thank you for giving your time to explain it to me. It makes kind of sense to me now. And you are right. As said I don't have a math background at all. The last time I had math in school was more then 10 years ago. I am thinking to take some tutoring in basic mathematical formulas. Though, going through the answers I had wrong in the GMAT Diagnostic test, when I understand the explanation of the writer but instead translate it into my own logic and language I tend to understand it even better. So there is still hope for me haha

I am a Arts student. I began with multimedia and graduated from University in American Studies. Yes not exactly Quantitative. However the Verbal and Integrated Reasoning sections of GMAT should and I found so far much easier for me.

So thanks again. I will continue to revise and take some Math classes here in the UK.

Ted

Ted,

I'll also suggest that Magoosh could be very helpful for your Quantitative side. First of all, if nothing else, take advantage of our free blog. Here's a recent blog article:
https://magoosh.com/gmat/2012/fractions-on-the-gmat/
from there, you should be able to access everything on the blog.

Also, here's a free Magoosh question:
https://gmat.magoosh.com/questions/979
When you submit your answer, the following page will have a complete video explanation. Each one of our 800+ GMAT practice questions has its own video explanation, for accelerated learning. We also have over 150 Math Lessons, covering everything you need for GMAT Quant, from the most basic to the most advanced. Frankly, I think it's impossible to find a complete package of high quality GMAT prep for any cheaper.

Let me know if you have any further questions.

Mike

Here's a step-by-step approach 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"):


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

This question type is VERY COMMON on the GMAT, so be sure to master the technique.

RELATED VIDEO


MORE PRACTICE

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

1) Brand A

2) No Brand A

3) Brand B

4) No Brand B

We are given that of 200 households surveyed, 80 did not use either brand and 60 used only Brand A. We are also given that for every household that used both brands, 3 used only Brand B. Thus, we can let x = the number of households that used both brands and 3x = the number of households that only used Brand B. We need to determine how many households used both brands.

Let’s fill all of this into our table.



We can create the following equation with the “Total” column and determine x:

4x + 140 = 200

4x = 60

x = 15

Thus, 15 households used both brands of soap.

Alternative solution:

This is an overlapping set question. We can use the following formula:

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

Can you please explain "if x used both A and B, then 3x used only B (but not A).So, \(4x+140=200\) --> \(x=15\)"
I can't comprehend this part. it is specifically mentioned in the question that 3 used 'only' brand B soap. So, why it is dependable on "both brands of soap" part. if so, why isn't 60 written as "60x".
help.

"For every household that used both brands of soap, 3 used only Brand B soap" means that if x used both A and B, then 3x used only B (but not A).

So, for example:
If 10 people used both brands of soap, 3*10 = 30 used only Brand B soap.
If 12 people used both brands of soap, 3*12 = 36 used only Brand B soap.
If x people used both brands of soap, 3x used only Brand B soap.

Solution:

Given:

• Number of households surveyed = 200

• Households who used only Brand A soaps = 60

• Households who used only Brand B soaps = 3* Households who used both soaps.

• Households who used neither Brand A nor Brand B soaps = 80

Working out:

We need to find out the number of households which used soaps of both the brands.

To understand the solution, let us draw a 2-set Venn diagram



Since the total households surveyed were 200 and 80 of them used no soaps, (200-80) households used at least one of Brand A or Brand B soaps.

• Thus, \(3x +x +60 = 120\)

• Or, \(x =15\).

Thus, 15 households used both the brands of the soaps.

Answer: Option A