To increase the sales by 30%, the fare of the ticket was reduced by 20%. What was the percentage change in revenue?

Let's solve this step by step.

Given:
- The sales (quantity of tickets sold) increased by 30%.
- The fare of the ticket was reduced by 20%.

### Step 1: Define the initial values.
Let's assume:
- The initial number of tickets sold is \( Q \).
- The initial fare per ticket is \( P \).

Thus, the initial revenue \( R_{\text{initial}} \) is:
\[
R_{\text{initial}} = Q \times P
\]

### Step 2: Calculate the new number of tickets sold.
Since sales increased by 30%, the new number of tickets sold \( Q_{\text{new}} \) is:
\[
Q_{\text{new}} = Q \times (1 + 0.30) = 1.3Q
\]

### Step 3: Calculate the new fare per ticket.
The fare was reduced by 20%, so the new fare per ticket \( P_{\text{new}} \) is:
\[
P_{\text{new}} = P \times (1 - 0.20) = 0.8P
\]

### Step 4: Calculate the new revenue.
The new revenue \( R_{\text{new}} \) is:
\[
R_{\text{new}} = Q_{\text{new}} \times P_{\text{new}} = 1.3Q \times 0.8P = 1.04QP
\]

### Step 5: Determine the percentage change in revenue.
The percentage change in revenue is calculated as:
\[
\text{Percentage change in revenue} = \frac{R_{\text{new}} - R_{\text{initial}}}{R_{\text{initial}}} \times 100\%
\]

Substituting the values:
\[
\text{Percentage change in revenue} = \frac{1.04QP - QP}{QP} \times 100\% = \frac{0.04QP}{QP} \times 100\% = 4\%
\]

### Final Answer:
The percentage change in revenue is **4%**.

Do you want any more details or have any questions?

Here are 8 related questions to consider:
1. What would be the revenue change if the fare was reduced by 30% instead of 20%?
2. How would a 40% increase in sales affect revenue if the ticket fare remained constant?
3. What if both sales and fare decreased by 20%, how would the revenue change?
4. How do you calculate percentage change in a business context?
5. If the fare reduction resulted in only a 10% increase in sales, what would be the revenue change?
6. What other factors could influence the change in revenue besides ticket price and quantity sold?
7. How would the break-even point change if costs increased by 10% while prices were reduced?
8. Can you derive a formula for revenue change based on percentage changes in price and quantity?

**Tip:** When analyzing percentage changes, consider how changes in one variable (like price) might influence other variables (like quantity sold) and their combined effect on overall outcomes (like revenue).

Learn how to calculate the percentage change in revenue when sales increase by 30% and ticket fare is reduced by 20%. Understand the formula for percentage change and its application in business scenarios. Suitable for high school students and beyond.

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