How many people in a group of 50 own neither a fax machine nor a laser

Total = T=50(given)
Fax Owners= F
Laser Printer Owners = L
Both Owners=B
Neither = N

T=F+L-B+N
st1) We just know N#0. INSUF
St2) B=15.--> 50=F+L-15+N; F+L could be anything. INSUF
1+2) we still need F+L INSUF

Ans E

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 independent equations ensures a solution.

How many people in a group of 50 own neither a fax machine nor a laser printer?

(1) The total number of people in the group who own a fax machine or a laser printer or both is less than 50.
(2) The total number of people in the group who own both a fax machine and a laser printer is 15.

When it comes to a question like the above, the question is frequently given in the Gmat Math test, which is "2 by 2" que like the table below.

When you look at the table, there are 4 variables(a,b,c,d) and 1 equation(a+b+c+d=50), which should match with the number of equations. So, you need 3 more equations. For 1) 1 equation, For 2) 1 equation, which is likely to make E the answer. In 1) & 2), it is a+b+c<50, a=15 and the value of d is not unique, which is not sufficient. Therefore, the answer is E.

-> For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

(1) The total of Fax U Laser could be any number <= 50. So, total owning neither could be any number <= 50. Hence, not sufficient.
(2) Total of Fax intersection Laser is 15. However, consider no(fax only) = 16 and no(laser only) = 18. This will give no(neither) = 50 - 49=1. On the other hand, consider that no(fax only) = 0 (possible, since all those who own a fax machine could own a laser as well) and no(laser only) = 10. In that case, no(neither) = 25. Not sufficient.

With (1) and (2) together, we still don't know the distribution. (1) doesn't really give any additional info. for (2) since in a group of 50, (1) is bound to be true, except for the exception where it is exactly 50, whether it is mentioned or not. So, both together reduces to (2) only (if you consider that we can get the =50 case, but we can also get a large number of other cases where both (1) and (2) hold true), which is insufficient.

So, (E) - BOTH TOGETHER ARE INSUFFICIENT.

We see that we have an overlapping set problem with two categories:

1) Own a fax machine

2) Own a laser printer

We can also create a few variables.

F = total number of people who own a fax machine

L = total number of people who own a laser printer

B = number of people who own both a laser printer and a fax machine

N = number of people who own neither a laser printer nor a fax machine

We are given that the group consists of 50 people. Thus, we can create the following equation:

50 = F + L – B + N

Note that we subtract B in the equation because those who own both a laser printer and a fax machine were double-counted, once in F and again in L.

We must determine the value of N.

Statement One Alone:

The total number of people in the group who own a fax machine or a laser printer or both is less than 50.

Using the information in statement one, we can create the following inequality:

F + L – B < 50

We see that we can determine that N > 0; however, we cannot determine the value of N. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

The total number of people in the group who own both a fax machine and a laser printer is 15.

Using the information in statement two, we know that B = 15. This is not enough information to determine the value of N. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two, and the given information, we know the following:

50 = F + L – B + N

F + L – B < 50

B = 15

We see that this is not enough information to determine the value of N.

Answer: E
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I know its very silly doubt but plz clear this:
(1) The total number of people in the group who own a fax machine or a laser printer or both is less than 50.
as per diagram 1, in attacheed image,
should it be
(a+c)[fax owners]+(b+c)[laser]+c [both]<50

or it is :
a[only fax]+ b[only laser] + c[both], 50
if it is only fax case then why only fax is not mentioned?
Say I am given total 60 trucks ,40 have airbags, 25 have power steering and 12 have both,
here airbags =a+c not only a?
then why this discrepancy?
plz clear

Please, someone, clear the question with double matrix method.

Total = n(Fax) + n (Laser) - n(Both) + n(Neither)

n(Fax or Laser) = n(Fax) + n(Laser) - n(Both)

From question stem, Total = 50

50 = n(Fax or Laser) + n(Neither)

Stmnt 1: n(Fax or Laser) < 50. This just tells us that there are at least some people who own neither. We don't know how many.

Stmnt 2: n(Both) = 15

Using both statements, we cannot say how many lie in n(Fax or Laser). Hence, we cannot find the value of n(Neither).

Answer (E)
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Answer: Option E

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