Visitors to a show were charged ₹ $15$ each on the first day, ₹ $7.50$ each on the second day and ₹ $2.50$ each 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 & Formula This problem requires finding the weighted average. When exact numbers aren't given but ratios are, you can use the ratio components directly as the "weights" or assumed number of visitors, because the common multiplier will cancel out in the fraction. $$ \text{Weighted Average} = \frac{w_1x_1 + w_2x_2 + w_3x_3}{w_1 + w_2 + w_3} $$ Step-by-Step Solution * Given charges for Day 1, Day 2, and Day 3: ₹ $15$, ₹ $7.50$, and ₹ $2.50$. * Given attendance ratio: $2 : 5 : 13$. Let the attendance be $2x$, $5x$, and $13x$. * Total number of visitors over three days = $2x + 5x + 13x = 20x$. * Calculate total revenue collected: - Day 1: $2x \times 15 = 30x$ - Day 2: $5x \times 7.50 = 37.5x$ - Day 3: $13x \times 2.50 = 32.5x$ * Total revenue = $30x + 37.5x + 32.5x = 100x$. * Average charge per person = $\frac{\text{Total Revenue}}{\text{Total Visitors}} = \frac{100x}{20x} = 5$. Exam Strategy & Shortcut Ignore the variable $x$ completely in exams. Assume exactly $2$, $5$, and $13$ people visited. Calculate the total money collected: $(2 \times 15) + (5 \times 7.5) + (13 \times 2.5) = 30 + 37.5 + 32.5 = 100$. Divide by the total number of people ($2 + 5 + 13 = 20$). So, $100 / 20 = 5$. Common Pitfall A common error is just taking the average of the three ticket prices: $(15 + 7.5 + 2.5) / 3 = 25 / 3 = 8.33$. This is conceptually wrong because it assumes an equal number of people visited each day, totally disregarding the $2:5:13$ attendance ratio. Final Answer **Therefore, the correct answer is ₹ 5.**