Last year in a group of 30 businesses, 21 reported a net profit and 15

Last year in a group of 30 businesses, 21 reported a net profit and 15 had investments in foreign markets. How many of the businesses did not report a net profit nor invest in foreign markets last year?

Given: {Total} = {Profit} + {Investment} - {Both} + {Neither} --> 30=21+15-{Both}+{Neither} --> {Neither} = {Both} - 6.
Question: {Neither} = ?

(1) Last year 12 of the 30 businesses reported a net profit and had investments in foreign markets --> {Both} = 12 --> {Neither}={Both}-6=12-6=6. Sufficient.

(2) Last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both --> {Profit} + {Investment} - {Both} = 24 --> 21+15-{Both}=24 --> {Both}=12 --> {Neither}={Both}-6=12-6=6. Sufficient.

Answer: D.


As you can see the problem can be easily solved without a matrix, but anyway here you go, matrix for statement (2):

(2) says that a+b+c=24, on the other hand a+b+c+d=30=Total --> 24+d=30 --> d=6. Sufficient.

Hope it's clear.

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Last year in a group of 30 businesses, 21 reported a net profit and 15 had investments in foreign markets. How many of the businesses did not report a net profit nor invest in foreign markets last year?

(1) Last year 12 of the 30 businesses reported a net profit and had investments in foreign markets.
(2) Last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both.


Lets make a table to understand the given information


We need to find x

St1:profit and investment were 12 companies so we can find about companies with investment but nonprofit which is 3 so 3+x=9 or x=6 so sufficient


St2 means that 24 companies had shown either profit or investment or both so 30-24=6 companies with no profit and no investment
T= Only A(Profit)+ OnlyB(Investment)+ Both(investment and profit) +Neither and thus Neither= 6
Ans is D
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I do not understand how can we say "last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both" when the question says "Last year in a group of 30 businesses, 21 reported a net profit and 15 had investments in foreign markets."

Isn't B going against the question?

Statement in question : Last year in a group of 30 businesses, 21 reported a net profit
Statement B : last year 24 of the 30 businesses reported a net profit

Refer to the matrix:

a = investment and profit;
b = investment and NO profit;
c = profit and NO investment;
d = NO investment and NO profit.

(2) Last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both --> a+b+c=24 (all groups but NO investment and NO profit.).

Hope it's clear.
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Presenting the Venn diagram approach of the solution

Interpreting the given info
We are told about 30 businesses of which 21 reported a net profit and 15 had investments in foreign markets.Refer the venn diagram below:



As the total number of businesses is 30 we can write a + b + c + d = 30

Also, 21 firms reported a net profit, that would give us a + b =21. Also, it would give us c + d = 9

Similarly, 15 firms had investments in foreign markets, that would give us b + c = 15. Also it would give us a + d = 15.
We need to find the number of businesses that did not report a net profit nor invest in foreign market i.e. d. Let's see if the statements provide us the information to find a definite value of d

Statement-I
St-I tells us that 12 businesses reported net profit and had investments in foreign markets i.e. b = 12. This would give us the value of a = 9.
We know that a + d = 15 i.e. d = 6.
Hence st-I is sufficient to answer the question.

Statement-II
St-II tells us that 24 businesses reported a net profit or invested in foreign markets or both i.e. a + b + c = 24

Since a + b + c + d = 30 we can find the value of d = 6

Hence st-II is sufficient to answer the question.

Answer is option D

Hope this helps

Regards
Harsh
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The Double Matrix Method can be used for most questions featuring a population in which each member has two characteristics associated with it.

Here, we have a population of businesses, and the two characteristics are:
- had a net profit or didn't have a net profit
- invested in foreign markets or didn't invest in foreign markets

So, we can set up our diagram as follows:


Now, if there are 30 businesses and 21 reported a net profit, then 9 did NOT report a net profit.
Similarly, if there are 30 businesses and 15 had investments in foreign markets, then 15 did NOT have investments in foreign markets. So, we can add that information to the diagram.

Note: I placed a star in the bottom right box to remind me that this is the value we are trying to determine.

Statement 1: Last year 12 of the 30 businesses reported a net profit and had investments in foreign markets.
When we add this information to our diagram, we get the following:


Since we know the sums of the rows and columns, we can determine the values in every box to get the following:

Notice that we can determine the value for the bottom right box, which means we can answer the target question with certainty.
So, statement 1 is SUFFICIENT

Statement 2: last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both.
The three shaded boxes below represent the businesses that reported a net profit or invested in foreign markets, or both.


Since all 4 boxes must add to 30, we can see last remaining box must contain the 6 remaining businesses.

So, we can determine the value for the bottom right box, which means statement 2 is SUFFICIENT

Answer: D

RELATED VIDEO

(2) is really saying that there were 6 of 30 (30-24=6) that did not invest in foreign markets or make a profit.

From that, it is subtraction to solve that rest.

Bunuel

When i read S2 -- per the table -- i thought the equation was 3a + b + c = 24

Here is why when i read S2

Last year 24 of the 30 businesses reported a
net profit ===================================================== this is a + c as net profit is ACROSS column 1
or invested in foreign markets ========================== this is a + b as investing in foreign markets is ACROSS row 1
or
both ========================================================= this is a as this relates to one cell one

Given the or, lets add of them up thus

- a + c + a + b + a, which is 3a + b + c

Where is the gap in my logic

Hi EgmatQuantExpert

When i read S2 -- per the Venn Diagram -- i thought the equation for S2 was 3b + a + c = 24 (not a + b + c = 24)

Here is why when i read S2

Last year 24 of the 30 businesses reported a
net profit ===================================================== this is a + b as net profit includes the shared region as well
or invested in foreign markets ========================== this is b+c as investing in foreign markets is ACROSS the shared region as well
or
both ========================================================= this is "b" as this relates to one cell one

Given the or, lets add of them up thus

- a + b + c + b + b, which is 3b + a + c

How am i thinking through this incorrectly ?

Hi karishma

Remembered your post regarding another overlapping set problem when thinking about S2 in this problem

Shouldn't S2 in this problem be actually saying :

Last year 24 of the 30 businesses reported a net profit ONLY or invested in foreign markets ONLY or both.


Karishma's post on the significance of 'ONLY' : https://gmatclub.com/forum/of-the-84-parents-who-attended-a-meeting-at-a-school-135191.html#p1199542

Without the "ONLY" -- there is clear overlap, is there not

Last year 24 of the 30 businesses reported a
net profit ===================================================== this is a + b as net profit includes the shared region as well
or invested in foreign markets ======================================== this is b+c as investing in foreign markets is ACROSS the shared region as well
or
both ======================================================== this is "b" as this relates to one cell one


Given the "or", lets add of them up thus

- a + b + c + b + b, which is 3b + a + c = 24

How am i thinking through this incorrectly ?

Bunuel can you please help me with the formulation of ST 2?

Total - (Neither/Nor) = Profit + Investment - Both
30 - (N) = 21 + 15 - (B)

Hi EgmatQuantExpert

When i read statement two specifically -- i thought per the Venn Diagram, it was 3b + a + c instead of a+b+c

Last year 24 of the 30 businesses reported a
net profit ===================================================== this is a + b as net profit is ENTIRE circle 1
or invested in foreign markets ========================== this is b + c as investing in foreign markets is ENTIRE circle 2
or
both ========================================================= this is "B" to the overlap

hence

a+b+b+c+b = 3b + a + c

Please let me know where is the gap in my logic

Hey Bunuel

Is it really required for the statement 2 to go like the way you described. Can we do directly in this way ?

Total- None = Union.
We know Union is 24 as per statement II and as per original question stem we know Total = 30.
So None = 30-24 = 6.
Do we have to find the intersection really or am i missing something ?

Thanks
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OFFICIAL GMAT EXPLANATION

Arithmetic Concepts of sets

Consider the Venn diagram below in which x represents the number of businesses that reported a net profit and had investments in foreign markets. Since 21 businesses reported a net profit, 21 − x businesses reported a net profit only. Since 15 businesses had investments in foreign markets, 15 − x businesses had investments in foreign markets only. Finally, since there is a total of 30 businesses, the number of businesses that did not report a net profit and did not invest in foreign markets is 30 − (21− x + x + 15 − x) = x − 6.

Determine the value of x − 6, or equivalently, the value of x.

The image contains a Venn diagram with two circles; the one on the left represents profit and the one on the right represents foreign. Inside the circle marked as “Profit”, there is the text “21 - x”, and inside the circle marked as “Foreign”, there is the text “15 - x.” The region where both these circles overlap contains the text “x.” Below the Venn diagram is the text “x - 6.”

1. It is given that 12 = x; SUFFICIENT.
2. It is given that 24 = (21 − x) + x + (15 − x). Therefore, 24 = 36 − x, or x = 12.

Alternatively, the information given is exactly the number of businesses that are not among those to be counted in answering the question posed in the problem, and therefore the number of businesses that are to be counted is 30 − 24 = 6; SUFFICIENT.

Each statement alone is sufficient.

Solution:

We need to determine the number of businesses that did not report a net profit nor invest in foreign markets last year. If we let x be this number and y be the number of businesses that both report a net profit and invest in foreign markets last year, we can create the equation:

30 = 21 + 15 - y + x

y - 6 = x

Notice that we need to determine the value of x; but if we know the value of y, then we can determine the value of x.

Statement One Alone:

We see that y = 12, thus x = 12 - 6 = 6. Statement one alone is sufficient.

Statement Two Alone:

We can create the equation:

21 + 15 - y = 24

y = 12

Since y = 12, x = 12 - 6 = 6. Statement two alone is sufficient.

Answer: D

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