Two-thirds of a consignment was sold at a profit of 6% and the rest at a loss of 3%. If there was an overall profit of Rs 540, find the value of the consigment.

Answer

To solve the problem step by step, we need to find the total value of the consignment based on the given conditions. Let's break it down:

### Step 1: Define the Variables
Let the total cost price (CP) of the consignment be \( x \).

### Step 2: Determine the Portions Sold
According to the problem, two-thirds of the consignment was sold at a profit of 6%, and one-third was sold at a loss of 3%.

- Cost price of two-thirds of the consignment:
\[ \text{CP of } \frac{2}{3} = \frac{2}{3}x \]

- Cost price of one-third of the consignment:
\[ \text{CP of } \frac{1}{3} = \frac{1}{3}x \]

### Step 3: Calculate Selling Prices
1. **Selling Price (SP) of two-thirds sold at 6% profit:**
\[ \text{SP} = \text{CP} + \text{Profit} = \frac{2}{3}x + 0.06 \times \frac{2}{3}x = \frac{2}{3}x \times (1 + 0.06) = \frac{2}{3}x \times 1.06 = \frac{2.12}{3}x \]

2. **Selling Price (SP) of one-third sold at 3% loss:**
\[ \text{SP} = \text{CP} - \text{Loss} = \frac{1}{3}x - 0.03 \times \frac{1}{3}x = \frac{1}{3}x \times (1 - 0.03) = \frac{1}{3}x \times 0.97 = \frac{0.97}{3}x \]

### Step 4: Calculate Total Selling Price
Now, we can find the total selling price (SP) of the entire consignment:
\[ \text{Total SP} = \text{SP of two-thirds} + \text{SP of one-third} \]
\[ \text{Total SP} = \frac{2.12}{3}x + \frac{0.97}{3}x = \frac{(2.12 + 0.97)}{3}x = \frac{3.09}{3}x \]

### Step 5: Calculate Overall Profit
According to the problem, the overall profit is Rs 540. The overall profit can also be expressed as:
\[ \text{Overall Profit} = \text{Total SP} - \text{Total CP} \]
Substituting the values we have:
\[ 540 = \frac{3.09}{3}x - x \]

### Step 6: Simplify the Equation
To simplify:
\[ 540 = \frac{3.09 - 3}{3}x = \frac{0.09}{3}x \]
Multiplying both sides by 3:
\[ 1620 = 0.09x \]
Now, divide both sides by 0.09:
\[ x = \frac{1620}{0.09} = 18000 \]

### Step 7: Conclusion
The value of the consignment is Rs 18,000.