If a shopkeeper marks an item `50%` above its CP and discount is given on the marked price and `12%` discount is given on the market price and the shopkeeper makes profit of rs `256`, what will be the actual cost price of the item? (in rs)

Options

A. 1000
B. 800
C. 750
D. 1200

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define the Cost Price (CP)
Let the actual cost price (CP) of the item be Rs. 100 (this is a convenient assumption to simplify calculations).

### Step 2: Calculate the Marked Price (MP)
Since the item is marked 50% above its cost price:
\[ \text{Marked Price (MP)} = \text{CP} + 50\% \text{ of CP} = 100 + 0.5 \times 100 = 100 + 50 = Rs. 150 \]

### Step 3: Calculate the Selling Price (SP)
A discount of 12% is given on the marked price. Therefore, the selling price can be calculated as:
\[ \text{Selling Price (SP)} = \text{MP} - 12\% \text{ of MP} = 150 - 0.12 \times 150 \]
Calculating the discount:
\[ 0.12 \times 150 = 18 \]
Thus,
\[ \text{SP} = 150 - 18 = Rs. 132 \]

### Step 4: Relate Selling Price to Cost Price and Profit
We know that the profit is given as Rs. 256. The profit can be expressed as:
\[ \text{Profit} = \text{SP} - \text{CP} \]
Substituting the values we have:
\[ 256 = 132 - \text{CP} \]

### Step 5: Solve for Cost Price (CP)
Rearranging the equation gives:
\[ \text{CP} = 132 - 256 \]
Calculating the cost price:
\[ \text{CP} = 132 - 256 = -124 \quad \text{(This indicates an error in our assumption)} \]

### Step 6: Correct the Approach
Instead of assuming CP as Rs. 100, we will denote it as \( x \):
1. Marked Price (MP) = \( x + 0.5x = 1.5x \)
2. Selling Price (SP) = \( 1.5x - 0.12 \times 1.5x = 1.5x \times 0.88 = 1.32x \)

### Step 7: Set Up the Profit Equation
The profit is given as Rs. 256:
\[ \text{Profit} = \text{SP} - \text{CP} = 1.32x - x = 256 \]
This simplifies to:
\[ 0.32x = 256 \]

### Step 8: Solve for \( x \)
Dividing both sides by 0.32:
\[ x = \frac{256}{0.32} = 800 \]

### Conclusion
The actual cost price of the item is Rs. 800.

---