A man sells 3 types of articles at 10%, 20% and 30% profit respectively. If the ratio of the cost of the articles is 1 : 2 : 3 and the ratio of the number of articles of each type sold is 3 : 2 : 1, then what is his net profit?

### Concept & Weighted Average The net profit percentage on multiple types of items can be found by calculating the total profit generated and dividing it by the total cost price incurred. $$ \text{Net Profit \%} = \left( \frac{\text{Total Profit}}{\text{Total Cost}} \right) \times 100 $$ ### Step-by-Step Solution 1. Let the cost price (CP) of the 3 types of articles be $1x, 2x, \text{ and } 3x$. 2. Let the number of articles sold of each type be $3y, 2y, \text{ and } 1y$. 3. Calculate the Total Cost (Total CP): - Total CP of Type 1 = $1x \times 3y = 3xy$ - Total CP of Type 2 = $2x \times 2y = 4xy$ - Total CP of Type 3 = $3x \times 1y = 3xy$ - Total CP = $3xy + 4xy + 3xy = 10xy$ 4. Calculate the Total Profit: - Profit on Type 1 (10%) = $10\% \text{ of } 3xy = 0.3xy$ - Profit on Type 2 (20%) = $20\% \text{ of } 4xy = 0.8xy$ - Profit on Type 3 (30%) = $30\% \text{ of } 3xy = 0.9xy$ - Total Profit = $0.3xy + 0.8xy + 0.9xy = 2.0xy$ 5. Calculate the Net Profit Percentage: $$ \text{Net Profit \%} = \left( \frac{2.0xy}{10xy} \right) \times 100 = 20\% $$ ### Exam Strategy & Shortcut Assume convenient numbers instead of variables. Let costs be ₹100, ₹200, ₹300 and quantities be 3, 2, 1. Total cost = $300 + 400 + 300 = 1000$. Total profit = $10\%$ of 300 + $20\%$ of 400 + $30\%$ of 300 = $30 + 80 + 90 = 200$. Net Profit % = $\frac{200}{1000} \times 100 = 20\%$. ### Common Pitfall Simply averaging the profit percentages $(10\% + 20\% + 30\%) / 3 = 20\%$. Although this gives the correct numerical answer by pure coincidence in this specific problem (due to symmetrical weighting), this method is fundamentally flawed and will fail in most other scenarios. Always use weighted sums. ### Final Answer Therefore, the correct answer is **20%**.