If each of the 20 bolts of fabric on a shelf is either 100 percent cot

Let, 100 Cotton = C
100 Wool = W
Mixture of cotton and wool = M

Given, C+W+M = 20

Question : M=?

Statement 1: Of the 20 bolts, 18 contain some wool and 14 contain some cotton.
i.e. W+M = 18 and C+M = 14
we know, C+W+M = 20
i.e. C = 2, W=6 and M= 12
SUFFICIENT

Statement 2:Of the 20 bolts, 6 are 100 percent wool.
W=6
i.e. C+M=14
NOT SUFFICIENT

Answer: option A

n(W)=Bolts containing Wool
n(C)=Bolts containing Cotton
n(Total)= Total number of bolts
n(C and W)=Bolts containing Wool and Cotton


From A:
\(n(W)=18\),\(n(C)=14\), \(n(Total)=20\) ,\(n(C and W)=x\)

\(20= 18+14-x\) . Solving for x gives \(x=12\)

Option A is sufficient

From B:
\(n(W)=6\)

However there is no information about other type of bolts . B is insufficient

Correct answer is Option A

W: only wool
C: Only cotton
M: Mix

W+C+M = 20
(1) Of the 20 bolts, 18 contain some wool and 14 contain some cotton.
W+C+2M = 32
=> Know the value of M
=> SUFFICIENT
(2) Of the 20 bolts, 6 are 100 percent wool.
Just know W
=> Cannot know M
=> INSUFFICIENT

Ans: A

In sets questions, statements often give us more information than they let on.

(1) Of the 20 bolts, 18 contain some wool and 14 contain some cotton.
18 contain some wool means not only "18 contain at least some wool", but also "2 contain no wool". So 2 must have only cotton.
14 contain some cotton means not only "14 contain at least some cotton", but also "6 contain no cotton". So 6 have only wool.

20 = Only wool + Only cotton + Both = 6 + 2 + Both
Both = 12
Sufficient

(2) Of the 20 bolts, 6 are 100 percent wool.
6 contain only wool. We don't know how many contain only cotton.
Not sufficient.

Answer (A)
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Hi GMATinsight,
Although I solved the problem like you approach, I would like to know whether I can solve it using double matrix approach. I tried but failed.

Thanks

Yes it can be solved with Double Matrix approach as well.

Here is the solution

Thanks. i found my mistake I missed '0' in No wool and No cotton.

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.

If each of the 20 bolts of fabric on a shelf is either 100 percent cotton, 100 percent wool, or a mixture of cotton and wool, how many bolts contain both cotton and wool?

(1) Of the 20 bolts, 18 contain some wool and 14 contain some cotton.
(2) Of the 20 bolts, 6 are 100 percent wool.

This is a typical ‘2 by 2 question’, a commonly observed type of question in GMAT math. We can represent the original condition using the table below:

There is no fabric that is neither cotton nor wool, so this can be written ‘0’, and a+b+c=20,
Giving 3 variables (a,b,c) and an equation. We need 2 more equations in order to solve for the variables, and 2 equations are given from the 2 conditions, making (C) likely to be the answer.
However, looking at condition 1,
a+b=18, a+c=14 (2 equations), making the condition sufficient,
and condition 2 gives b=6, being insufficient, makes the answer (A).

VERITAS PREP OFFICIAL SOLUTION:

Question Type: What Is the Value? This question asks for number of bolts that contain BOTH cotton and wool.

Given information in the question stem or diagram: 20 bolts of fabric total; each is 100% cotton, or 100% wool, or a mixture of BOTH cotton and wool. This means that neither = 0 and you can set up your Venn diagram equation before you go to the statements: Total = Cotton (C) + Wool (W) – Both (B) or 20 = Cotton + Wool – Both. Remember that C and W represent the total sets not those with “Only Wool” or “Only Cotton.”

Statement 1: Of the 20 bolts, 18 contain some wool and 14 contain some cotton. Plugging this information into the equation above you see that this statement is sufficient. 20 = 18 + 14 – B. B = 12. Or logically you know that there are 20 total bolts but 18 + 14 = 32. This means the difference must be the number of bolts that are double counted (those with both). The answer is either A or D.

Statement 2: Of the 20 bolts 6 are 100% wool. As in any Venn Diagram problem, when you are given “only” information such as this, you must use a diagram or think logically (the equation above is for total set information, not “only” information). If “Wool Only” = 6 then you know that the total of cotton bolts (including those with both) is 14. However there is no way to determine the value of “Both” from this information so it is not sufficient. The correct answer is A. Note: Almost all Venn diagram problems hinge on reading carefully and distinguishing between “Total Set” information and “Only” information. In this problem, it is the difference between bolts containing Wool or Cotton (which includes both) and bolts containing 100% Wool or Cotton or Only Wool or Cotton (which does not include both).
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I opted for option A as the anwer by applying the traditional theory of sets i.e.,

But I wonder whether the wordings some wool and some cotton are appropriate in meaning?
I felt they are adding meaning wise confusion.
Can someone please comment? Engr2012

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