If `2/3` part of an article is sold at `30%` profit, `1/4` part of `16%` proft and remaining part at 12% profit and finally, there is a profit of Rs. 75. then find the cost price of the article.

Answer

Let's solve the problem step by step.

### Step 1: Define the Cost Price
Let the cost price of the article be \( CP \).

### Step 2: Calculate the Selling Price for Each Part
1. **For \( \frac{2}{3} \) of the article sold at 30% profit:**
- Selling Price (SP) for this part:
\[ SP_1 = \frac{2}{3} \times CP \times \left(1 + \frac{30}{100}\right) = \frac{2}{3} \times CP \times 1.3 = \frac{2.6}{3} \times CP \]

2. **For \( \frac{1}{4} \) of the article sold at 16% profit:**
- Selling Price (SP) for this part:
\[ SP_2 = \frac{1}{4} \times CP \times \left(1 + \frac{16}{100}\right) = \frac{1}{4} \times CP \times 1.16 = \frac{1.16}{4} \times CP \]

3. **For the remaining part:**
- The remaining part of the article is:
\[ 1 - \left(\frac{2}{3} + \frac{1}{4}\right) = 1 - \left(\frac{8}{12} + \frac{3}{12}\right) = 1 - \frac{11}{12} = \frac{1}{12} \]
- Selling Price (SP) for this part at 12% profit:
\[ SP_3 = \frac{1}{12} \times CP \times \left(1 + \frac{12}{100}\right) = \frac{1}{12} \times CP \times 1.12 = \frac{1.12}{12} \times CP \]

### Step 3: Calculate Total Selling Price
Now, we can find the total selling price (SP) of the article:
\[ SP = SP_1 + SP_2 + SP_3 \]
Substituting the values we calculated:
\[ SP = \frac{2.6}{3} \times CP + \frac{1.16}{4} \times CP + \frac{1.12}{12} \times CP \]

### Step 4: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple of 3, 4, and 12 is 12. So we convert each term:
\[ SP = \left(\frac{2.6 \times 4}{12} + \frac{1.16 \times 3}{12} + \frac{1.12}{12}\right) \times CP \]
Calculating each term:
- \( \frac{2.6 \times 4}{12} = \frac{10.4}{12} \)
- \( \frac{1.16 \times 3}{12} = \frac{3.48}{12} \)
- \( \frac{1.12}{12} = \frac{1.12}{12} \)

Now, adding these:
\[ SP = \frac{10.4 + 3.48 + 1.12}{12} \times CP = \frac{14.0}{12} \times CP = \frac{7}{6} \times CP \]

### Step 5: Calculate Profit
The profit is given as Rs. 75. Profit can also be calculated as:
\[ Profit = SP - CP \]
Substituting the SP we found:
\[ 75 = \frac{7}{6} \times CP - CP \]
This simplifies to:
\[ 75 = \left(\frac{7}{6} - 1\right) \times CP = \left(\frac{7}{6} - \frac{6}{6}\right) \times CP = \frac{1}{6} \times CP \]

### Step 6: Solve for Cost Price
Now, we can solve for \( CP \):
\[ CP = 75 \times 6 = 450 \]

### Final Answer
The cost price of the article is \( \text{Rs. } 450 \).

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