M27-04

Official Solution:


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

This is a tricky question.

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

If we denote the number of astronauts who do NOT listen to Bach as \(x\), then the number of male astronauts who do NOT listen to Bach can be represented as \(0.56x\). It is important to note that \(0.56x\) can be expressed as \(\frac{14}{25}x\), and it needs to be an integer. Consequently, \(x\) must be a multiple of 25, such as 25, 50, 75, and so on. Additionally, since \(x\) cannot exceed the total number of astronauts, which is 35, the only feasible value for \(x\) is 25. Therefore, the number of astronauts who do listen to Bach is given by subtracting \(x\) from the total number of astronauts: \(35 - 25 = 10\). Sufficient..

(2) Of the astronauts who listen to Bach, 70% are female.

Applying the same logic as before, let's denote the number of astronauts who listen to Bach as \(y\). In this case, the number of female astronauts who listen to Bach can be expressed as \(0.7y\), which is equivalent to \(\frac{7}{10}y\). To ensure this value is an integer, \(y\) must be a multiple of 10. However, unlike the previous scenario, there are multiple possible values that satisfy this condition: 10, 20, and 30. Therefore, this statement alone is not sufficient to determine the exact number of astronauts who listen to Bach.


Answer: A