Question: If at least one astronaut does NOT listen to Bach at Solaris space station, then how many of 35 astronauts at Solaris space station listen to Bach?

(1) Of the astronauts who do NOT listen to Bach, 56% are male.
(2) Of the astronauts who listen to Bach, 70% are female.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: A
Solution and Explanation:
Approach Solution 1:
The problem statement states that:

Given:

• At least one astronaut does NOT listen to Bach at Solaris space station.

Find out:

• How many of the 35 astronauts at Solaris space station listen to Bach.

Statement 1 alone: Of the astronauts who do NOT listen to Bach, 56% are male.
If the number of astronauts who do NOT listen to Bach is x, then 0.56x is the number of males who do NOT listen to Bach.
It is required to notice that 0.56x= 14/25 x must be an integer.
Hence, x must be a multiple of 25: 25, 50, 75, …
But x (number of astronauts who do NOT listen to Bach) must also be less than (or equal to) 35.
So x can only be 25, which makes the number of astronauts who do listen to Bach equal to 35−25=10.
Hence, statement 1 alone is Sufficient.

Statement 2 alone: Of the astronauts who listen to Bach, 70% are female.
Now, if we apply the same logic here we get that if the number of astronauts who listen to Bach is y, then 0.7y is the number of females who listen to Bach.
It is required to note that 0.7y= 7/10 y must be an integer.
Hence, it must be a multiple of 10, but in this case, it can take more than 1 value: 10, 20, 30.
Therefore, this statement alone is not sufficient.

Approach Solution 2:
The problem statement states that:

Given:

• At least one astronaut does NOT listen to Bach at Solaris space station.

Find out:

• How many of the 35 astronauts at Solaris space station listen to Bach.

Statement 1: Of the astronauts who do NOT listen to Bach, 56% are male.
Statement 2: Of the astronauts who listen to Bach, 70% are female

We can solve the problem by using a double-set matrix.
After analysing the data of statement 1 and statement 2, we get:

Let’s consider statement 1:
From statement 1 we get, of the astronauts who do NOT listen to Bach, male = 0.56x and female = 0.44x
Both 0.56x and 0.44x must be integers.
To make 0.56 an integer, we need to convert it to a fraction first.
0.56 = 56/100 = 14/25
Therefore, by multiplying it by 25, we will get an integer.
Therefore, x (least value) = 25 (>1).
Then, y = 35 - 25 = 10

Next Possible value of x = 25*2 (least multiple of 25) > 35
Hence, there is only one possible value of x.
Therefore, statement 1 alone is Sufficient.

Let’s consider statement 2:
From statement 2 we get, of the astronauts who listen to Bach, female = 0.7y and male = 0.3y.
Both 0.7y and 0.3y must be integers.
To make 0.7 an integer, we need to convert it to a fraction first.
0.7 = 7/10
Therefore, the minimum value of y will be 10 to make the number an integer.
The next possible value of y will be 10*2 = 20.
If y = 10, then x = 35 - 10 = 25
If y = 20, then x = 35 - 20 = 15
Therefore, we are more than one value.
Hence, statement 2 alone is not sufficient.

“If at least one astronaut does NOT listen to Bach at Solaris space station”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions are featured with a problem statement that is followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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