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At a community arts center with 240 members, every member attended at least one of two programs: studio classes or outdoor workshops. If 25 percent of the members who attended studio classes also attended outdoor workshops, did more than 50 percent of the members attend studio classes?
(1) Fewer than 60 percent of the members attended outdoor workshops.
(2) More than 90 members attended only studio classes.
(1) Fewer than 60 percent of the members attended outdoor workshops.
(2) More than 90 members attended only studio classes.
30 May 2026, 09:59
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Official Solution:
At a community arts center with 240 members, every member attended at least one of two programs: studio classes or outdoor workshops. If 25 percent of the members who attended studio classes also attended outdoor workshops, did more than 50 percent of the members attend studio classes?
\(\{both\} = \frac{1}{4} * \{studio\ classes\}\)
So:
\(\{only\ studio\ classes\} = \{studio\ classes\} - \{both\}\)
\(\{only\ studio\ classes\} = \{studio\ classes\} - \frac{1}{4} * \{studio\ classes\}\)
\(\{studio\ classes\} = \frac{4}{3} * \{only\ studio\ classes\}\)
We need to know whether more than 50 percent of the 240 members attended studio classes. That means we need to know whether
\(\{studio\ classes\} > 120\)
Since \(\{studio\ classes\} = \frac{4}{3} * \{only\ studio\ classes\}\), this is equivalent to asking whether
\(\frac{4}{3} * \{only\ studio\ classes\} > 120\)
\(\{only\ studio\ classes\} > 90\)
(1) Fewer than 60 percent of the members attended outdoor workshops.
So fewer than 144 members attended outdoor workshops. Since every member attended at least one of the two programs,
\(\{Total\} = \{only\ studio\ classes\} + \{outdoor\ workshops\}\)
\(\{only\ studio\ classes\} = \{Total\} - \{outdoor\ workshops\}\)
\(\{only\ studio\ classes\} = 240 - \{outdoor\ workshops\}\)
Since \(\{outdoor\ workshops\} < 144\), we get
\(\{only\ studio\ classes\} > 96\)
That is enough to conclude that \(\{only\ studio\ classes\} > 90\), so \(\{studio\ classes\} > 120\).
Sufficient.
(2) More than 90 members attended only studio classes.
This gives \(\{only\ studio\ classes\} > 90\), which is exactly what we need to know. Sufficient.
Answer: D
At a community arts center with 240 members, every member attended at least one of two programs: studio classes or outdoor workshops. If 25 percent of the members who attended studio classes also attended outdoor workshops, did more than 50 percent of the members attend studio classes?
\(\{both\} = \frac{1}{4} * \{studio\ classes\}\)
So:
\(\{only\ studio\ classes\} = \{studio\ classes\} - \{both\}\)
\(\{only\ studio\ classes\} = \{studio\ classes\} - \frac{1}{4} * \{studio\ classes\}\)
\(\{studio\ classes\} = \frac{4}{3} * \{only\ studio\ classes\}\)
We need to know whether more than 50 percent of the 240 members attended studio classes. That means we need to know whether
\(\{studio\ classes\} > 120\)
Since \(\{studio\ classes\} = \frac{4}{3} * \{only\ studio\ classes\}\), this is equivalent to asking whether
\(\frac{4}{3} * \{only\ studio\ classes\} > 120\)
\(\{only\ studio\ classes\} > 90\)
(1) Fewer than 60 percent of the members attended outdoor workshops.
So fewer than 144 members attended outdoor workshops. Since every member attended at least one of the two programs,
\(\{Total\} = \{only\ studio\ classes\} + \{outdoor\ workshops\}\)
\(\{only\ studio\ classes\} = \{Total\} - \{outdoor\ workshops\}\)
\(\{only\ studio\ classes\} = 240 - \{outdoor\ workshops\}\)
Since \(\{outdoor\ workshops\} < 144\), we get
\(\{only\ studio\ classes\} > 96\)
That is enough to conclude that \(\{only\ studio\ classes\} > 90\), so \(\{studio\ classes\} > 120\).
Sufficient.
(2) More than 90 members attended only studio classes.
This gives \(\{only\ studio\ classes\} > 90\), which is exactly what we need to know. Sufficient.
Answer: D
