Retailer sold one article at `33(1)/(3)%` profit and another at 100% profit. Find 'his overall profit percentage if the selling price of both the article is same?

Options

A. A)`60%`
B. B)`55%`
C. C)`66(2)/(3)%`
D. D)`75%`

Answer

To solve the problem step by step, we will first analyze the profit percentages and then calculate the overall profit percentage based on the selling prices being the same.

### Step 1: Understand the Profit Percentages
The retailer sold one article at a profit of \(33 \frac{1}{3}\%\) and another at \(100\%\).

### Step 2: Convert Profit Percentages to Fractions
- \(33 \frac{1}{3}\%\) can be converted to a fraction:
\[ 33 \frac{1}{3}\% = \frac{100}{3}\% = \frac{1}{3} \]
- \(100\%\) is simply \(1\).

### Step 3: Set Up the Cost Prices
Let’s assume the cost price (CP) of the first article is \(CP_1\) and the second article is \(CP_2\).

- For the first article:
\[ SP_1 = CP_1 + \text{Profit} = CP_1 + \frac{1}{3} CP_1 = \frac{4}{3} CP_1 \]

- For the second article:
\[ SP_2 = CP_2 + \text{Profit} = CP_2 + 1 \cdot CP_2 = 2 CP_2 \]

### Step 4: Set Selling Prices Equal
Since the selling prices of both articles are the same, we can set them equal:
\[ \frac{4}{3} CP_1 = 2 CP_2 \]

### Step 5: Express Cost Prices in Terms of Each Other
From the equation above:
\[ CP_1 = \frac{3}{8} \cdot 2 CP_2 = \frac{3}{4} CP_2 \]

### Step 6: Calculate Total Cost Price
Now, we can find the total cost price:
\[ \text{Total CP} = CP_1 + CP_2 = \frac{3}{4} CP_2 + CP_2 = \frac{3}{4} CP_2 + \frac{4}{4} CP_2 = \frac{7}{4} CP_2 \]

### Step 7: Calculate Total Selling Price
The total selling price is:
\[ \text{Total SP} = SP_1 + SP_2 = \frac{4}{3} CP_1 + 2 CP_2 \]
Substituting \(CP_1\):
\[ = \frac{4}{3} \left(\frac{3}{4} CP_2\right) + 2 CP_2 = CP_2 + 2 CP_2 = 3 CP_2 \]

### Step 8: Calculate Overall Profit
Profit can be calculated as:
\[ \text{Profit} = \text{Total SP} - \text{Total CP} = 3 CP_2 - \frac{7}{4} CP_2 = \left(3 - \frac{7}{4}\right) CP_2 = \left(\frac{12}{4} - \frac{7}{4}\right) CP_2 = \frac{5}{4} CP_2 \]

### Step 9: Calculate Overall Profit Percentage
Profit percentage is given by:
\[ \text{Profit Percentage} = \left(\frac{\text{Profit}}{\text{Total CP}}\right) \times 100 = \left(\frac{\frac{5}{4} CP_2}{\frac{7}{4} CP_2}\right) \times 100 \]
This simplifies to:
\[ = \left(\frac{5}{7}\right) \times 100 = \frac{500}{7} \approx 71.43\% \]

### Step 10: Conclusion
Thus, the overall profit percentage is approximately \(71.43\%\).