1/3rd of the total of an article is sold at a profit of 15%, 1/4th is at a profit of 20% and the remained part is at a profit of 24%. If the total profit is 6,240 find the cost price of the article.

Options

A. Rs 35000
B. Rs 31000
C. Rs 32000
D. Rs 31200

Answer

To solve the problem step by step, we will find the cost price of the article based on the given profits from different fractions of the article sold.

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

### Step 2: Determine the Parts Sold
1. One-third of the article is sold, which is \( \frac{1}{3} \) of \( CP \).
2. One-fourth of the article is sold, which is \( \frac{1}{4} \) of \( CP \).
3. The remaining part is sold, which we will calculate next.

### Step 3: Calculate the Remaining Part
To find the remaining part, we first need to calculate the total fraction sold:
- Total fraction sold = \( \frac{1}{3} + \frac{1}{4} \)

Finding a common denominator (which is 12):
- \( \frac{1}{3} = \frac{4}{12} \)
- \( \frac{1}{4} = \frac{3}{12} \)

So,
\[ \frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \]

Thus, the remaining part is:
\[ 1 - \frac{7}{12} = \frac{5}{12} \]

### Step 4: Calculate the Profit from Each Part
1. Profit from the first part (\( \frac{1}{3} \)):
- Amount sold = \( \frac{1}{3} CP \)
- Profit = 15% of \( \frac{1}{3} CP = 0.15 \times \frac{1}{3} CP = \frac{0.15 CP}{3} = \frac{0.15 CP}{3} \)

2. Profit from the second part (\( \frac{1}{4} \)):
- Amount sold = \( \frac{1}{4} CP \)
- Profit = 20% of \( \frac{1}{4} CP = 0.20 \times \frac{1}{4} CP = \frac{0.20 CP}{4} = \frac{0.20 CP}{4} \)

3. Profit from the remaining part (\( \frac{5}{12} \)):
- Amount sold = \( \frac{5}{12} CP \)
- Profit = 24% of \( \frac{5}{12} CP = 0.24 \times \frac{5}{12} CP = \frac{0.24 \times 5}{12} CP = \frac{1.2 CP}{12} \)

### Step 5: Total Profit Calculation
Now, we sum up all the profits:
\[ \text{Total Profit} = \left( \frac{0.15 CP}{3} + \frac{0.20 CP}{4} + \frac{1.2 CP}{12} \right) \]

Finding a common denominator (which is 12):
- \( \frac{0.15 CP}{3} = \frac{0.15 \times 4}{12} CP = \frac{0.6 CP}{12} \)
- \( \frac{0.20 CP}{4} = \frac{0.20 \times 3}{12} CP = \frac{0.6 CP}{12} \)

Thus,
\[ \text{Total Profit} = \left( \frac{0.6 CP}{12} + \frac{0.6 CP}{12} + \frac{1.2 CP}{12} \right) = \frac{2.4 CP}{12} = \frac{0.2 CP}{1} \]

### Step 6: Set Up the Equation
Given that the total profit is 6,240, we can set up the equation:
\[ 0.2 CP = 6240 \]

### Step 7: Solve for Cost Price
To find \( CP \):
\[ CP = \frac{6240}{0.2} = 31200 \]

### Conclusion
The cost price of the article is \( \text{CP} = 31,200 \).