A manufacturer conducted a survey to determine how many peop

I believe, this is the quickest possible solution.

Cheers!

A manufacturer conducted a survey to determine how many people buy products P and Q. What fraction of the people surveyed said that they buy neither product P nor product Q?

You can solve this question with Venn diagram, matrix or as shown below.

{Total} = {buy P} + {buy Q} - {buy both P and Q} + {buy neither P nor Q}.
Question: {buy neither P nor Q} / {Total} = ?

Take total to be equal to 6 (as it's a multiple of both 2 and 3)

(1) 1/3 of the people surveyed said that they buy product P but not product Q:

{buy P} - {buy both P and Q} = 1/3*6 = 2;
6 = {buy P} + {buy Q} - ({buy P} - 2) + {buy neither P nor Q}
4={buy Q} + {buy neither P nor Q}.

Not sufficient to get the ratio we need.

(2) 1/2 of the people surveyed said that they buy product Q:

{buy Q}=1/2*6=3. Not sufficient.

(1)+(2) 4={buy Q} + {buy neither P nor Q} and {buy Q} = 3;
{buy neither P nor Q} = 1;
{buy neither P nor Q}/{Total} = 1/6. Sufficient.

Answer: C.
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Venn diagram makes this question MUCH easier:First about the formula: {Total} = {buy P} + {buy Q} - {buy both P and Q} + {buy neither P nor Q}. Note that P={Only P}+{Both P&Q} and Q={Only Q}+{Both P&Q}. In {Total}={P}+{Q}-{Both P&Q}+{Neither P nor Q} we subtract {Both P&Q} as P and Q both contain this segment and thus in P+Q it's counted twice, so we should subtract it to count it only once.

Back to the question:

Again let's take total to be equal to 6: so 6=Green+Yellow+Blue+Grey. We need to get Grey/Total=Grey/6=?.

(1) 1/3 of the people surveyed said that they buy product P but not product Q --> Green=1/3*6=2. Not sufficient to get the ratio we need.

(2) 1/2 of the people surveyed said that they buy product Q --> Yellow+Blue=1/2*6=3. Not sufficient.

(1)+(2) 6=Green+Yellow+Blue+Grey=2+3+Grey --> Grey=1 --> Grey/Total=1/6. Sufficient.

Answer: C.

Hope it's clear.
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This problem should be solved following way

There are 2 products "P" and "Q", and we have to answer what fraction of people do not select both the products.

Option 1- 1/3 of people select only P and not Q
So, suppose we have 90 people responded to survey then 1/3 of 90 = 30 people select only product P
But, this option does not tells us anything about Q, so not sufficient to answer the question.

Option 2- 1/2 people select product Q ....this includes people who selected inly product Q and people who selected
both product P and Q i.e. P intersection Q
Therefore if 90 people responded to survey 45 people selected product Q, but it does not tells us how
many people select only product p

Now, it we combine the options it gives us value of A U B i.e if we have 90 people on the survey 30 selected product P and 45 selected product Q along with P intersection Q

Therefore P U Q = 30 + 45 = 75
Therefore (P U Q)' = 90 - 75 = 15
Hence fraction of people did not select any product = 15/90

Hence we get the answer by taking both the options together... hence answer "C"

i still dont understand.

lets take ur choices:

1. the total is 6. (1) means we have 2 ppl buy product P no Q.
(2) means 3 ppl buy Q (mayb together with P also).
i still cannot understand how u can figure out from that the area covered by both P+Q and by
none. Im not sure what im missing, but as i see it, we can have ppl that buy both P+Q
between none to 4 and it still wont make any logic problem with both sentences.

I guess im having hard time to understand why u chose to put the - {buy both P and Q}
in minus and not plus. the total should be a sum of all groups together isnt it?

thanks a lot for all the time and help for both of u guys.

Here's the equation:

True # of objects = (everyone in group 1) + (everyone in group 2) - (everyone in both groups) + people in neither group

You ask why we subtract everyone in both groups; it's because we've already counted those people twice!

If we break down the first two components

everyone in group 1 = (people only in group 1) + (people in both groups)
everyone in group 2 = (people only in group 2) + (people in both groups)

you can see that we've counted "people in both groups" twice. Subbing into the original equation:

True # of objects = ((people only in group 1) + (people in both groups)) + ((people only in group 2) + (people in both groups)) - (everyone in both groups) + people in neither group

which is why we need to subtract "everyone in both groups" to end up only counting them once.

Thanks for the explanations guys.

Bunuel - thanks a lot for the time u put in the drawings and all.

appreciated very much.

+1 for both. thanks.

Since we the objective is to obtain the value of 1 - PUQ there is a simple solution to this problem using the following formula:

PUQ = P + Q - PiQ

Statement 1 provides the following information:
P = 1/3 + PiQ

Statement 2 provides the following information:
Q = 1/2

Then we can conclude that
PUQ = 1/3 + PiQ + 1/2 - PiQ
PUQ = 5/6

Thus non-buyers are 1 - PUQ = 1 - 5/6 = 1/6

BOTH STATEMENTS TOGETHER ARE SUFFICIENT

Hey Bunuel this isn't clear to me because you write
(1)+(2) 6=Green+Yellow+Blue+Grey.

Isn't it suppose to be 6=Green-Yellow+Blue+Grey?

To get the entire space (total) we should add Green, Yellow, Blue, and Grey areas.

About the difference check here: a-polling-company-found-that-of-300-households-surveyed-148727.html?hilit=diagram#p1191678

Hope it helps.
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With the Venn Diagram it makes sense but writing it out algabreically I wouldn't be able to do it I would get as far as this and wouldn't know how to figure out both the only Q and both values. I think I am missing a simple implication of the 2nd statement and what affect it has.


Total= P + Q -both +neither
6= 2+

Can you please tell me what didn't you understand in the explanation of the second statement here: a-manufacturer-conducted-a-survey-to-determine-how-many-peop-106092.html#p831244 or here:https://gmatclub.com/forum/a-manufacturer-conducted-a-survey-to-determine-how-many-peop-106092.html#p831338 ?
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i could solve the que with help of ven dia. but i am unable to understand the highlighted portion of explanation. pls help.

{Total} = {buy P} + {buy Q} - {buy both P and Q} + {buy neither P nor Q}.

From (1): {buy P} - {buy both P and Q} = 1/3*6 = 2, so {buy both P and Q} = {buy P} - 2.

Substitute in above: 6 = {buy P} + {buy Q} - ({buy P} - 2) + {buy neither P nor Q}
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Forget conventional ways of solving math questions. In DS, Variable approach is

the easiest and quickest way to find the answer without actually solving the

problem. Remember equal number of variables and equations ensures a solution.


A manufacturer conducted a survey to determine how many people buy products P and Q. What fraction of the people surveyed said that they buy neither product P nor product Q?

(1) 1/3 of the people surveyed said that they buy product P but not product Q.
(2) 1/2 of the people surveyed said that they buy product Q.

==> this is a common 2by2 question in GMAT tests that we use variable approach method to solve.

as we can see from above, the original condition is asking for d and since we have 4 variables (a,b,c,d), we need 4 equations to match the number of variables and equations. Since we have 1 each in 1) and 2), E is likely the answer. Using both 1) and 2) together, c=1/3 and a+b=1/2 thus we have c+d=1/2. substituting c=1/3 gives us d=1/6, therefore the answer is C. Here we were able to find the answer, but normally for 90% of these questions with 4 variables the the answer is E. This case was a special case.

This is a great candidate for a technique called the Double Matrix Method. It can be used for most questions featuring a population in which each member has two criteria associated with it.
Here, the criteria are:
- buy product Q or not buy product Q
- buy product P or not buy product P
When I scan the two statements I see that they mention 1/3 of the people surveyed and 1/2 of the people surveyed. I also see that the target question asks us to find a fraction of the people surveyed (not the actual number). So, let's pick a nice number that works well with 1/3 and 1/2.
Let's say there 12 people were surveyed.

So, we'll start by setting up our diagram like this:


We want to find the fraction of the people surveyed said that they neither buy product P nor product Q. So, to answer this question, we need to know the number of people in the bottom right box (where the happy face is).


We're now ready to check the statements.

Statement 1: 1/3 of the people surveyed said that they buy product P but not product Q.
1/3 of 12 = 4. So, 4 people buy product P but not product Q.

Does this provide enough information to find the number of people in the bottom right box (where the happy face is)?
No.
Statement 1 is NOT SUFFICIENT

Statement 2: 1/2 of the people surveyed said that they buy product Q.
So, 6 people buy product Q, which means 6 people do not buy product Q

Does this provide enough information to find the number of people in the bottom right box (where the happy face is)?
No.
Statement 2 is NOT SUFFICIENT


Statements 1 and 2 combined:
We get:

Does this provide enough information to find the number of people in the bottom right box (where the happy face is)?
YES.

Since the two boxed in the right-hand column must add to 6, the bottom right box must have 2 people, which means 2/12 people said that they neither buy product P nor product Q.

Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

RELATED VIDEO


MORE PRACTICE

Prompt analysis
Let the total be 1
Therefore 1 = n(neither P nor Q) +n(P only) + n (Q only) + n(P and Q)
Also n(P) = n(P only) + n(P and Q) and n(Q) = n (Q only) + n(P and Q)

Superset
The value of the n (neither P nor Q) will lie in the range 0-1.

Translation
In order to find the value, we need
1# exact value of n (neither P nor Q)
2# exact value of rest of the variables in equation in prompt analysis
3# collated summation of 2 or 3 variables.

Statement analysis

St 1: n(P only) = ⅓. No information about the rest of the variables.INSUFFICIENT
St 2: n(Q) = n (Q only) + n(P and Q) = ½. No information about n (P only) and n (neither P nor Q). INSUFFICIENT

St 1 & St 2: n(Q) = n (Q only) + n(P and Q) = ½ and n(P only) = ⅓. Therefore
n (neither P nor Q) = 1 -½ -⅓ = ⅙. ANSWER

Option C

I assumed the total number of products to be 300 and then used the Venn Diagram approach to arrive at the final answer.

Is this a correct way to approach such problems? Bunuel

A manufacturer conducted a survey to determine how many people buy products P and Q. What fraction of the people surveyed said that they buy neither product P nor product Q?

Total no of People surveyed, U= n(P) + n(Q) - n(P and Q) + n( no P and no Q)

n(P) = no of people who buy Product P
n(Q) = no of people who buy Product Q
n(P and Q) = no of people who buy both Product P and Q
n(no P and no Q) = no of people who buy neither Product P nor Q

Here ,we need to find n(no P and no Q) as a fraction of total people surveyed.

(1) 1/3 of the people surveyed said that they buy product P but not product Q.

Here, its given that no of people who buy only product P i.e n(only P) = 1/3 * Total no of people surveyed = 1/3*U
Since there is no info regarding n(Q), we will not be able to find n( no P and no Q).
Statement 1 alone is insufficient


(2) 1/2 of the people surveyed said that they buy product Q.
n(Q) = 1/2* U.
We don't have any information about n(only P). Statement 2 alone is insufficient

Combining statement 1 and 2 :
n(only P) = 1/3*U
n(Q) = 1/2* U.

U= n(P) + n(Q) - n(P and Q) + n( no P and no Q)

n(only P)= n(P)- n(P and Q)

U= n(only P) + n(Q)+ n( no P and no Q)
U = 1/3*U + 1/2*U + n( no P and no Q)
n( no P and no Q) = U(1- 1/3 - 1/2) = 1/6*U

Option C is the correct answer.

Thanks,
Clifin J Francis,
GMAT SME
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