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A concert hall sold only two types of tickets for a show: Premium and Standard. Was the revenue from Premium tickets greater than 75 percent of the concert hall’s total ticket revenue?
(1) Each Premium ticket cost more than twice as much as each Standard ticket cost.
(2) The hall sold 4 Premium tickets for every 3 Standard tickets.
(1) Each Premium ticket cost more than twice as much as each Standard ticket cost.
(2) The hall sold 4 Premium tickets for every 3 Standard tickets.
26 Apr 2026, 13:17
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Official Solution:
A concert hall sold only two types of tickets for a show: Premium and Standard. Was the revenue from Premium tickets greater than 75 percent of the concert hall’s total ticket revenue?
Let \(p\) be the price of a Premium ticket and \(s\) be the price of a Standard ticket.
Let \(m\) be the number of Premium tickets sold and \(n\) be the number of Standard tickets sold.
We are asked whether Premium revenue is greater than 75 percent of total revenue:
\(m * p > 0.75(m * p + n * s)\)
\(mp > 3ns\)
So we need to know whether \(\frac{mp}{ns} > 3\).
(1) Each Premium ticket cost more than twice as much as each Standard ticket cost.
This gives \(\frac{p}{s} > 2\), but \(\frac{m}{n}\) is unknown, so \(\frac{mp}{ns}\) is unknown. Not sufficient.
(2) The hall sold 4 Premium tickets for every 3 Standard tickets.
This gives \(\frac{m}{n} = \frac{4}{3}\), but \(\frac{p}{s}\) is unknown, so \(\frac{mp}{ns}\) is unknown. Not sufficient.
(1) + (2) From (2), \(\frac{m}{n} = \frac{4}{3}\), so the question becomes:
Is \(\frac{mp}{ns} > 3\)
Is \(\frac{p}{s} * \frac{m}{n} > 3\)
Is \(\frac{p}{s} * \frac{4}{3} > 3\)
Is \(\frac{p}{s} > \frac{9}{4}\)
So the question becomes: do we know that \(\frac{p}{s} > \frac{9}{4}\)?
From (1), we only know \(\frac{p}{s} > 2\). If \(\frac{p}{s}\) is between 2 and \(\frac{9}{4}\), the answer is No, but if \(\frac{p}{s}\) is greater than \(\frac{9}{4}\), the answer is Yes. Not sufficient.
Answer: E
A concert hall sold only two types of tickets for a show: Premium and Standard. Was the revenue from Premium tickets greater than 75 percent of the concert hall’s total ticket revenue?
Let \(p\) be the price of a Premium ticket and \(s\) be the price of a Standard ticket.
Let \(m\) be the number of Premium tickets sold and \(n\) be the number of Standard tickets sold.
We are asked whether Premium revenue is greater than 75 percent of total revenue:
\(m * p > 0.75(m * p + n * s)\)
\(mp > 3ns\)
So we need to know whether \(\frac{mp}{ns} > 3\).
(1) Each Premium ticket cost more than twice as much as each Standard ticket cost.
This gives \(\frac{p}{s} > 2\), but \(\frac{m}{n}\) is unknown, so \(\frac{mp}{ns}\) is unknown. Not sufficient.
(2) The hall sold 4 Premium tickets for every 3 Standard tickets.
This gives \(\frac{m}{n} = \frac{4}{3}\), but \(\frac{p}{s}\) is unknown, so \(\frac{mp}{ns}\) is unknown. Not sufficient.
(1) + (2) From (2), \(\frac{m}{n} = \frac{4}{3}\), so the question becomes:
Is \(\frac{mp}{ns} > 3\)
Is \(\frac{p}{s} * \frac{m}{n} > 3\)
Is \(\frac{p}{s} * \frac{4}{3} > 3\)
Is \(\frac{p}{s} > \frac{9}{4}\)
So the question becomes: do we know that \(\frac{p}{s} > \frac{9}{4}\)?
From (1), we only know \(\frac{p}{s} > 2\). If \(\frac{p}{s}\) is between 2 and \(\frac{9}{4}\), the answer is No, but if \(\frac{p}{s}\) is greater than \(\frac{9}{4}\), the answer is Yes. Not sufficient.
Answer: E
