Visitors to a show were charged ₹ 15 each on the first day, ₹ 7.50 on the second day and ₹ 2.50 on the third day. The attendance on the three days was in the ratio 2 : 5 : 13. The average charge per person for the whole show was

### Concept & Weighted Average The average charge per person is found by dividing the total revenue collected over all days by the total number of visitors. $$\text{Average} = \frac{\sum (\text{Price}_i \times \text{Weight}_i)}{\sum \text{Weight}_i}$$ ### Step-by-Step Solution 1. **Given Prices:** Day 1 = ₹ 15, Day 2 = ₹ 7.50, Day 3 = ₹ 2.50. 2. **Given Weights (Attendance):** Ratio is $2 : 5 : 13$. Let the number of visitors be $2x, 5x,$ and $13x$. 3. **Calculate Total Visitors:** Total visitors = $2x + 5x + 13x = 20x$. 4. **Calculate Total Revenue:** Revenue Day 1 = $15 \times 2x = 30x$ Revenue Day 2 = $7.50 \times 5x = 37.5x$ Revenue Day 3 = $2.50 \times 13x = 32.5x$ Total Revenue = $30x + 37.5x + 32.5x = 100x$. 5. **Calculate Average Charge:** Average = $\frac{\text{Total Revenue}}{\text{Total Visitors}} = \frac{100x}{20x} = 5$. ### Exam Strategy & Shortcut Since ratios represent exact proportions, you can completely ignore any multiplier variable like $x$. Treat the ratio numbers as the exact number of people. Calculation becomes simple mental math: $\frac{15(2) + 7.5(5) + 2.5(13)}{2 + 5 + 13} = \frac{30 + 37.5 + 32.5}{20} = \frac{100}{20} = 5$. ### Common Pitfall A common error is taking the simple average of the charges: $\frac{15 + 7.50 + 2.50}{3} = \frac{25}{3} = 8.33$, which completely ignores the fact that different amounts of people paid each rate. ### Final Answer Therefore, the correct answer is **₹ 5**.