`(1)/(3)` of a commodity is sold at 15% profit, `(1)/(4)` is sold at 20% profit and the rest at 24% profit. If a total profit of Rs 62 is earned, then find the value of the commodity

Answer

To solve the problem, we will break it down step by step.

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

### Step 2: Calculate the Amount Sold at Each Profit Percentage
1. **Amount sold at 15% profit**:
\[ \text{Quantity sold} = \frac{1}{3} \text{ of } x \]
\[ \text{Profit} = 15\% \text{ of } \left(\frac{1}{3}x\right) = \frac{15}{100} \times \frac{1}{3}x = \frac{15x}{300} = \frac{x}{20} \]

2. **Amount sold at 20% profit**:
\[ \text{Quantity sold} = \frac{1}{4} \text{ of } x \]
\[ \text{Profit} = 20\% \text{ of } \left(\frac{1}{4}x\right) = \frac{20}{100} \times \frac{1}{4}x = \frac{20x}{400} = \frac{x}{20} \]

3. **Amount sold at 24% profit**:
The remaining quantity sold is:
\[ \text{Remaining quantity} = x - \left(\frac{1}{3}x + \frac{1}{4}x\right) \]
To simplify, find a common denominator (12):
\[ \frac{1}{3}x = \frac{4}{12}x, \quad \frac{1}{4}x = \frac{3}{12}x \]
\[ \text{Remaining quantity} = x - \left(\frac{4}{12}x + \frac{3}{12}x\right) = x - \frac{7}{12}x = \frac{5}{12}x \]
\[ \text{Profit} = 24\% \text{ of } \left(\frac{5}{12}x\right) = \frac{24}{100} \times \frac{5}{12}x = \frac{120x}{1200} = \frac{x}{10} \]

### Step 3: Calculate Total Profit
Now, we can sum up the profits from all three portions:
\[ \text{Total Profit} = \frac{x}{20} + \frac{x}{20} + \frac{x}{10} \]
To add these fractions, find a common denominator (20):
\[ \frac{x}{20} + \frac{x}{20} + \frac{2x}{20} = \frac{4x}{20} = \frac{x}{5} \]

### Step 4: Set Up the Equation
According to the problem, the total profit is Rs 62:
\[ \frac{x}{5} = 62 \]

### Step 5: Solve for x
To find the value of \( x \):
\[ x = 62 \times 5 = 310 \]

### Conclusion
The value of the commodity is Rs 310.

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