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03 Dec 2024, 01:30
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
78% (02:37) correct
22%
(02:49)
wrong
based on 169
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Out of 50 people surveyed, 20 said they liked Rock, 23 said they liked Hip Hop and 27 said they liked Alternative. In addition, 7 people said they liked both Rock and Hip Hop, and of those 7, 3 said they liked Alternative as well. If 10 only liked Hip Hop, 14 liked only Alternative and all of the 50 surveyed people liked at least one of the three types of music, how many liked ONLY Rock?
A. 8
B. 9
C. 12
D.13
E. 17
A. 8
B. 9
C. 12
D.13
E. 17
03 Dec 2024, 03:56
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Answer : 9 people liked only Rock
GMAT-Club-Forum-v6q73dlw.png [ 26.52 KiB | Viewed 464 times ]
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GMAT-Club-Forum-v6q73dlw.png [ 26.52 KiB | Viewed 464 times ]
08 Dec 2024, 09:14
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\(\)
Solution:
Step 1: Define the given data:
- Total surveyed: \( 50 \)
- People liking Rock (\( R \)): \( |R| = 20 \)
- People liking Hip Hop (\( H \)): \( |H| = 23 \)
- People liking Alternative (\( A \)): \( |A| = 27 \)
- People liking both Rock and Hip Hop (\( R \cap H \)): \( |R \cap H| = 7 \)
- People liking all three (\( R \cap H \cap A \)): \( |R \cap H \cap A| = 3 \)
- People liking only Hip Hop: \( |H_{\text{only}}| = 10 \)
- People liking only Alternative: \( |A_{\text{only}}| = 14 \)
Step 2: Calculate overlapping groups:
People who like both Rock and Hip Hop but not Alternative:
\[
|R \cap H| - |R \cap H \cap A| = 7 - 3 = 4
\]
Step 3: Calculate people liking only Rock:
The total number of people liking Rock is \( |R| = 20 \). To find those who like only Rock, we subtract the people who like Rock and overlap with other groups:
\[
|R_{\text{only}}| = |R| - \left( |R \cap H| + |R \cap A| - |R \cap H \cap A| \right)
\]
Step 4: Calculate \( |R \cap A| \):
The total number of people liking Alternative is \( |A| = 27 \). Subtract those who like only Alternative and those who like all three:
\[
|R \cap A| = |A| - |A_{\text{only}}| - |R \cap H \cap A| = 27 - 14 - 3 = 10
\]
Step 5: Substitute into \( |R_{\text{only}}| \):
\[
|R_{\text{only}}| = 20 - \left( 7 + 10 - 3 \right) = 20 - 14 = 6
\]
Final Answer:
C. 6
Solution:
Step 1: Define the given data:
- Total surveyed: \( 50 \)
- People liking Rock (\( R \)): \( |R| = 20 \)
- People liking Hip Hop (\( H \)): \( |H| = 23 \)
- People liking Alternative (\( A \)): \( |A| = 27 \)
- People liking both Rock and Hip Hop (\( R \cap H \)): \( |R \cap H| = 7 \)
- People liking all three (\( R \cap H \cap A \)): \( |R \cap H \cap A| = 3 \)
- People liking only Hip Hop: \( |H_{\text{only}}| = 10 \)
- People liking only Alternative: \( |A_{\text{only}}| = 14 \)
Step 2: Calculate overlapping groups:
People who like both Rock and Hip Hop but not Alternative:
\[
|R \cap H| - |R \cap H \cap A| = 7 - 3 = 4
\]
Step 3: Calculate people liking only Rock:
The total number of people liking Rock is \( |R| = 20 \). To find those who like only Rock, we subtract the people who like Rock and overlap with other groups:
\[
|R_{\text{only}}| = |R| - \left( |R \cap H| + |R \cap A| - |R \cap H \cap A| \right)
\]
Step 4: Calculate \( |R \cap A| \):
The total number of people liking Alternative is \( |A| = 27 \). Subtract those who like only Alternative and those who like all three:
\[
|R \cap A| = |A| - |A_{\text{only}}| - |R \cap H \cap A| = 27 - 14 - 3 = 10
\]
Step 5: Substitute into \( |R_{\text{only}}| \):
\[
|R_{\text{only}}| = 20 - \left( 7 + 10 - 3 \right) = 20 - 14 = 6
\]
Final Answer:
C. 6
