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26 Apr 2026, 14:28
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Last Saturday, the Aurora Theatre sold two types of tickets for the same show: Front Row tickets and Upper Level tickets. Each Front Row ticket sold for the same price, and each Upper Level ticket sold for the same price. What was the ratio of the theatre’s total revenue from Front Row tickets to its total revenue from Upper Level tickets?
(1) Last Saturday, the theatre sold 3 times as many Front Row tickets as Upper Level tickets.
(2) Last Saturday, each Upper Level ticket cost 20 euros more than each Front Row ticket.
(1) Last Saturday, the theatre sold 3 times as many Front Row tickets as Upper Level tickets.
(2) Last Saturday, each Upper Level ticket cost 20 euros more than each Front Row ticket.
26 Apr 2026, 14:28
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Official Solution:
Last Saturday, the Aurora Theatre sold two types of tickets for the same show: Front Row tickets and Upper Level tickets. Each Front Row ticket sold for the same price, and each Upper Level ticket sold for the same price. What was the ratio of the theatre’s total revenue from Front Row tickets to its total revenue from Upper Level tickets?
Let \(x\) be the price of a Front Row ticket and \(y\) be the price of an Upper Level ticket.
Let \(a\) be the number of Front Row tickets sold and \(b\) be the number of Upper Level tickets sold.
We want to find the value of \(\frac{xa}{yb}\).
(1) Last Saturday, the theatre sold 3 times as many Front Row tickets as Upper Level tickets.
This gives \(a = 3b\), so the ratio becomes \(\frac{x * 3b}{y * b} = \frac{3x}{y}\). Not sufficient.
(2) Last Saturday, each Upper Level ticket cost 20 euros more than each Front Row ticket.
This gives \(y = x + 20\), so the ratio becomes \(\frac{xa}{(x + 20)b}\). Not sufficient.
(1) + (2) When taken together, the ratio becomes \(\frac{xa}{yb} = \frac{x * 3b}{(x + 20)b} = \frac{3x}{x + 20}\), which still depends on \(x\), so it is not determined. Not sufficient.
Answer: E
Last Saturday, the Aurora Theatre sold two types of tickets for the same show: Front Row tickets and Upper Level tickets. Each Front Row ticket sold for the same price, and each Upper Level ticket sold for the same price. What was the ratio of the theatre’s total revenue from Front Row tickets to its total revenue from Upper Level tickets?
Let \(x\) be the price of a Front Row ticket and \(y\) be the price of an Upper Level ticket.
Let \(a\) be the number of Front Row tickets sold and \(b\) be the number of Upper Level tickets sold.
We want to find the value of \(\frac{xa}{yb}\).
(1) Last Saturday, the theatre sold 3 times as many Front Row tickets as Upper Level tickets.
This gives \(a = 3b\), so the ratio becomes \(\frac{x * 3b}{y * b} = \frac{3x}{y}\). Not sufficient.
(2) Last Saturday, each Upper Level ticket cost 20 euros more than each Front Row ticket.
This gives \(y = x + 20\), so the ratio becomes \(\frac{xa}{(x + 20)b}\). Not sufficient.
(1) + (2) When taken together, the ratio becomes \(\frac{xa}{yb} = \frac{x * 3b}{(x + 20)b} = \frac{3x}{x + 20}\), which still depends on \(x\), so it is not determined. Not sufficient.
Answer: E
