Question

The ratio of selling prices of four articles is \(3 : 5 : 6 : 8\) and their profits are 20%, 25%, 20% and 25% respectively. What is the overall percentage of profit on selling the four articles?

• A. \(11 \%\)
• B. \(22\frac{11}{12} \%\)
• C. \(24 \%\)
• D. \(28\frac{11}{12} \%\)
• E. \(36 \%\)

Answer 1

The Correct Option is B

Step 1: Understand the problem.
We are given that the selling prices of four articles are in the ratio \( 3 : 5 : 6 : 8 \), and their respective profits are 20%, 25%, 20%, and 25%. We need to find the overall percentage of profit on selling the four articles.

Step 2: Define variables.
Let the cost price of the four articles be \( C_1, C_2, C_3, C_4 \) and the selling price of the four articles be \( S_1, S_2, S_3, S_4 \). We know that the selling price is related to the cost price by the formula: \[ S = C + \text{Profit} \] The profit percentage is given as: \[ \text{Profit percentage} = \frac{\text{Profit}}{\text{Cost price}} \times 100 \] Therefore, the selling price can be written as: \[ S_1 = C_1 \times (1 + 0.20) = 1.20 C_1 \] \[ S_2 = C_2 \times (1 + 0.25) = 1.25 C_2 \] \[ S_3 = C_3 \times (1 + 0.20) = 1.20 C_3 \] \[ S_4 = C_4 \times (1 + 0.25) = 1.25 C_4 \]

Step 3: Use the ratio of selling prices.
The selling prices are in the ratio \( 3 : 5 : 6 : 8 \), so we can write: \[ S_1 : S_2 : S_3 : S_4 = 3 : 5 : 6 : 8 \] Let the constant of proportionality be \( k \), so: \[ S_1 = 3k, \, S_2 = 5k, \, S_3 = 6k, \, S_4 = 8k \] Using the relations for the selling prices in terms of the cost prices, we get: \[ 1.20 C_1 = 3k, \quad 1.25 C_2 = 5k, \quad 1.20 C_3 = 6k, \quad 1.25 C_4 = 8k \] Solving for the cost prices: \[ C_1 = \frac{3k}{1.20} = 2.5k, \quad C_2 = \frac{5k}{1.25} = 4k, \quad C_3 = \frac{6k}{1.20} = 5k, \quad C_4 = \frac{8k}{1.25} = 6.4k \]

Step 4: Calculate the total cost price and total selling price.
The total cost price is the sum of the individual cost prices: \[ \text{Total cost price} = C_1 + C_2 + C_3 + C_4 = 2.5k + 4k + 5k + 6.4k = 17.9k \] The total selling price is the sum of the individual selling prices: \[ \text{Total selling price} = S_1 + S_2 + S_3 + S_4 = 3k + 5k + 6k + 8k = 22k \]

Step 5: Calculate the overall profit percentage.
The total profit is the total selling price minus the total cost price: \[ \text{Total profit} = 22k - 17.9k = 4.1k \] The overall profit percentage is: \[ \text{Profit percentage} = \frac{\text{Total profit}}{\text{Total cost price}} \times 100 = \frac{4.1k}{17.9k} \times 100 = \frac{4.1}{17.9} \times 100 \approx 22\frac{11}{12} \% \]

Step 6: Conclusion.
The overall percentage of profit on selling the four articles is \( 22\frac{11}{12} \% \).

Final Answer:
The correct answer is (B): \( 22\frac{11}{12} \% \).