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06 Aug 2026, 03:59
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At a community center’s weekend open house, 360 registered visitors had access to two optional sessions: a gardening workshop and a budgeting seminar. How many of the registered visitors attended both sessions?
(1) Of the 360 registered visitors, 140 attended the gardening workshop, and 180 attended the budgeting seminar.
(2) The number of registered visitors who attended neither session was 40 greater than the number who attended both sessions.
(1) Of the 360 registered visitors, 140 attended the gardening workshop, and 180 attended the budgeting seminar.
(2) The number of registered visitors who attended neither session was 40 greater than the number who attended both sessions.
06 Aug 2026, 03:59
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Official Solution:
At a community center’s weekend open house, 360 registered visitors had access to two optional sessions: a gardening workshop and a budgeting seminar. How many of the registered visitors attended both sessions?
Let \(x\) be the number of registered visitors who attended both sessions.
(1) Of the 360 registered visitors, 140 attended the gardening workshop, and 180 attended the budgeting seminar.
Using the overlapping sets formula:
\(\{\text{Gardening or Budgeting}\} = 140 + 180 - x = 320 - x\)
So:
\(\{\text{Neither}\} = 360 - (320 - x) = 40 + x\)
This does not determine \(x\).
Not sufficient.
(2) The number of registered visitors who attended neither session was 40 greater than the number who attended both sessions.
So:
\(\{\text{Neither}\} = x + 40\)
This does not determine \(x\).
Not sufficient.
(1)+(2) From statement (1), we already get:
\(\{\text{Neither}\} = x + 40\)
Statement (2) gives the same relationship, so it adds no new information.
For example, if \(x = 60\):
\(\{\text{Gardening only}\} = 140 - 60 = 80\)
\(\{\text{Budgeting only}\} = 180 - 60 = 120\)
\(\{\text{Neither}\} = 100\)
If \(x = 90\):
\(\{\text{Gardening only}\} = 140 - 90 = 50\)
\(\{\text{Budgeting only}\} = 180 - 90 = 90\)
\(\{\text{Neither}\} = 130\)
Both cases satisfy both statements, but they give different values of \(x\).
Not sufficient.
Answer: E
At a community center’s weekend open house, 360 registered visitors had access to two optional sessions: a gardening workshop and a budgeting seminar. How many of the registered visitors attended both sessions?
Let \(x\) be the number of registered visitors who attended both sessions.
(1) Of the 360 registered visitors, 140 attended the gardening workshop, and 180 attended the budgeting seminar.
Using the overlapping sets formula:
\(\{\text{Gardening or Budgeting}\} = 140 + 180 - x = 320 - x\)
So:
\(\{\text{Neither}\} = 360 - (320 - x) = 40 + x\)
This does not determine \(x\).
Not sufficient.
(2) The number of registered visitors who attended neither session was 40 greater than the number who attended both sessions.
So:
\(\{\text{Neither}\} = x + 40\)
This does not determine \(x\).
Not sufficient.
(1)+(2) From statement (1), we already get:
\(\{\text{Neither}\} = x + 40\)
Statement (2) gives the same relationship, so it adds no new information.
For example, if \(x = 60\):
\(\{\text{Gardening only}\} = 140 - 60 = 80\)
\(\{\text{Budgeting only}\} = 180 - 60 = 120\)
\(\{\text{Neither}\} = 100\)
If \(x = 90\):
\(\{\text{Gardening only}\} = 140 - 90 = 50\)
\(\{\text{Budgeting only}\} = 180 - 90 = 90\)
\(\{\text{Neither}\} = 130\)
Both cases satisfy both statements, but they give different values of \(x\).
Not sufficient.
Answer: E
