To solve the problem, we need to analyze the information given about the members of the youth club and how they like different types of music. Let's break it down step-by-step. ### Step 1: Define Variables Let: - \( M \) = number of members who like modern music = 60 - \( T \) = number of members who like traditional music = 50 - \( B \) = number of members who like both types of music - \( N \) = number of members who do not like any type of music - \( U \) = number of members who like only modern music - \( V \) = number of members who like only traditional music ### Step 2: Use the Information Given From the problem, we know: 1. The total number of members in the club is 94. 2. The number of members who like both types of music is three times those who do not like any type of music: \( B = 3N \). ### Step 3: Set Up the Equations We can express the total number of members in terms of those who like modern music, traditional music, both, and neither: \[ U + V + B + N = 94 \] Where: - \( U = M - B \) (members who like only modern music) - \( V = T - B \) (members who like only traditional music) Substituting \( U \) and \( V \) into the total members equation gives: \[ (M - B) + (T - B) + B + N = 94 \] This simplifies to: \[ M + T - B + N = 94 \] ### Step 4: Substitute Known Values Now, substituting the known values of \( M \) and \( T \): \[ 60 + 50 - B + N = 94 \] This simplifies to: \[ 110 - B + N = 94 \] Rearranging gives: \[ N - B = 94 - 110 \] \[ N - B = -16 \quad \text{(1)} \] ### Step 5: Substitute \( B \) in Terms of \( N \) From the earlier information, we have \( B = 3N \). Substituting this into equation (1): \[ N - 3N = -16 \] \[ -2N = -16 \] \[ N = 8 \] ### Step 6: Find \( B \) Now that we have \( N \), we can find \( B \): \[ B = 3N = 3 \times 8 = 24 \] ### Step 7: Calculate \( U \) and \( V \) Now we can find \( U \) and \( V \): - \( U = M - B = 60 - 24 = 36 \) (members who like only modern music) - \( V = T - B = 50 - 24 = 26 \) (members who like only traditional music) ### Step 8: Calculate Members Who Like Only One Type of Music The total number of members who like only one type of music is: \[ U + V = 36 + 26 = 62 \] ### Conclusion Thus, the number of members who like only one type of music is **62**. ### Explanation of Options - **A. 8**: This is incorrect because it does not account for the members who like only one type of music. - **B. 24**: This is incorrect as it represents the number of members who like both types of music, not those who like only one type. - **C. 62**: This is the correct answer, as calculated above. - **D. 86**: This is incorrect because it exceeds the total number of members and does not reflect the distribution of music preferences. ### Revision Summary - Define variables for each group of members based on their music preferences. - Set up equations based on the total number of members and relationships between groups. - Solve for unknowns step-by-step, substituting back into equations as needed. - Verify calculations to ensure the final answer reflects the problem's requirements.