The number of seats in an auditorium is increased by 25%. The price of a ticket is also increased by 20%. What is the percentage increase in the revenue collected?

To determine the percentage increase in revenue, we need to consider both the increase in the number of seats and the increase in the ticket price.

Let's assume:
- The original number of seats in the auditorium is \( N \).
- The original price of a ticket is \( P \).

**Original Revenue**:
\[ \text{Original Revenue} = N \times P \]

**New Number of Seats**:
The number of seats is increased by 25%, so the new number of seats is:
\[ \text{New Number of Seats} = N \times (1 + 0.25) = N \times 1.25 \]

**New Price of a Ticket**:
The price of a ticket is increased by 20%, so the new price of a ticket is:
\[ \text{New Price of a Ticket} = P \times (1 + 0.20) = P \times 1.20 \]

**New Revenue**:
\[ \text{New Revenue} = (\text{New Number of Seats}) \times (\text{New Price of a Ticket}) \]
\[ \text{New Revenue} = (N \times 1.25) \times (P \times 1.20) \]
\[ \text{New Revenue} = N \times P \times 1.25 \times 1.20 \]

Simplifying the expression:
\[ \text{New Revenue} = N \times P \times 1.5 \]

**Percentage Increase in Revenue**:
The percentage increase in revenue is calculated as:
\[ \text{Percentage Increase} = \left(\frac{\text{New Revenue} - \text{Original Revenue}}{\text{Original Revenue}}\right) \times 100\% \]

Substituting the values:
\[ \text{Percentage Increase} = \left(\frac{N \times P \times 1.5 - N \times P}{N \times P}\right) \times 100\% \]
\[ \text{Percentage Increase} = \left(\frac{N \times P \times (1.5 - 1)}{N \times P}\right) \times 100\% \]
\[ \text{Percentage Increase} = \left(\frac{N \times P \times 0.5}{N \times P}\right) \times 100\% \]
\[ \text{Percentage Increase} = 0.5 \times 100\% \]
\[ \text{Percentage Increase} = 50\% \]

Thus, the percentage increase in the revenue collected is **50%**.

Would you like further details or have any questions?

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Here are five related questions to explore further:
1. What if the ticket price had been decreased by 10% instead? How would that affect revenue?
2. How would the result change if the number of seats had increased by 50% and the price remained the same?
3. What would be the new revenue if both the number of seats and ticket price were doubled?
4. Can you calculate the new revenue if only the number of seats increased but the ticket price remained constant?
5. How does the percentage increase in revenue compare if the price increase is higher than the seat increase?

**Tip**: In percentage increase problems, it's often helpful to set up equations that express new values as a function of the original values. This approach simplifies calculating changes in totals like revenue, cost, or quantity.

Learn how to calculate the percentage increase in revenue when the number of seats in an auditorium is increased by 25% and the ticket price is increased by 20%. This problem involves using formulas for percentage increase and revenue calculation. Suitable for high school students.

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