A shopkeeper sells `(3/4)`th of its articles at a gain of `20%` and the remaining at C.P. His real gain in the transaction is

Options

A. `10%`
B. `15%`
C. `20%`
D. `25%`

Answer

To solve the problem, we need to calculate the shopkeeper's real gain in the transaction step by step.

### Step 1: Define the Cost Price (C.P.)
Let's assume the total cost price of all articles is \( C \).

### Step 2: Calculate the Cost Price of Sold Articles
The shopkeeper sells \( \frac{3}{4} \) of the articles. Therefore, the cost price of the articles sold at a gain of 20% is:
\[ \text{C.P. of } \frac{3}{4} \text{ articles} = \frac{3}{4} C \]

### Step 3: Calculate the Selling Price (S.P.) of Sold Articles
The selling price of these articles, sold at a gain of 20%, can be calculated as:
\[ \text{S.P. of } \frac{3}{4} \text{ articles} = \text{C.P.} + \text{Gain} \]
Where Gain is 20% of C.P.:
\[ \text{Gain} = 20\% \text{ of } \frac{3}{4} C = \frac{20}{100} \times \frac{3}{4} C = \frac{3}{20} C \]
Thus,
\[ \text{S.P. of } \frac{3}{4} \text{ articles} = \frac{3}{4} C + \frac{3}{20} C \]

### Step 4: Find a Common Denominator
To add these fractions, we need a common denominator. The common denominator of 4 and 20 is 20. Therefore, we convert \( \frac{3}{4} C \) to a fraction with a denominator of 20:
\[ \frac{3}{4} C = \frac{15}{20} C \]
Now we can add:
\[ \text{S.P. of } \frac{3}{4} \text{ articles} = \frac{15}{20} C + \frac{3}{20} C = \frac{18}{20} C = \frac{9}{10} C \]

### Step 5: Calculate the Cost Price of Remaining Articles
The remaining articles, which is \( \frac{1}{4} \) of the total articles, are sold at cost price. Therefore, the cost price of these articles is:
\[ \text{C.P. of } \frac{1}{4} \text{ articles} = \frac{1}{4} C \]
And since they are sold at cost price, the selling price is also:
\[ \text{S.P. of } \frac{1}{4} \text{ articles} = \frac{1}{4} C \]

### Step 6: Calculate Total Selling Price
Now, we can find the total selling price of all articles:
\[ \text{Total S.P.} = \text{S.P. of } \frac{3}{4} \text{ articles} + \text{S.P. of } \frac{1}{4} \text{ articles} \]
\[ \text{Total S.P.} = \frac{9}{10} C + \frac{1}{4} C \]

### Step 7: Find a Common Denominator for Total S.P.
To add \( \frac{9}{10} C \) and \( \frac{1}{4} C \), we convert \( \frac{1}{4} C \) to a fraction with a denominator of 20:
\[ \frac{1}{4} C = \frac{5}{20} C \]
And convert \( \frac{9}{10} C \) to a fraction with a denominator of 20:
\[ \frac{9}{10} C = \frac{18}{20} C \]
Now we can add:
\[ \text{Total S.P.} = \frac{18}{20} C + \frac{5}{20} C = \frac{23}{20} C \]

### Step 8: Calculate the Real Gain
Now we can calculate the real gain:
\[ \text{Real Gain} = \text{Total S.P.} - \text{Total C.P.} \]
\[ \text{Real Gain} = \frac{23}{20} C - C = \frac{23}{20} C - \frac{20}{20} C = \frac{3}{20} C \]

### Step 9: Calculate the Gain Percentage
To find the gain percentage, we can use the formula:
\[ \text{Gain Percentage} = \left( \frac{\text{Real Gain}}{\text{Total C.P.}} \right) \times 100 \]
\[ \text{Gain Percentage} = \left( \frac{\frac{3}{20} C}{C} \right) \times 100 = \frac{3}{20} \times 100 = 15\% \]

### Final Answer
The shopkeeper's real gain in the transaction is **15%**.