An item is sold at two successive gains of `30%` and `20%.` If the final selling price is Rs. 31200, of the item what is the cost price (in Rs.) of the item ?

Options

A. 15000
B. 20000
C. 22250
D. 24000

Answer

To find the cost price (CP) of the item given the final selling price (SP) after two successive gains of 30% and 20%, we can follow these steps:

### Step 1: Understand the Selling Price Calculation
When an item is sold at successive gains, the selling price can be calculated using the formula:
\[ \text{SP} = \text{CP} \times (1 + \text{Gain}_1) \times (1 + \text{Gain}_2) \]
where \(\text{Gain}_1\) and \(\text{Gain}_2\) are the gains expressed as decimals.

### Step 2: Convert Percentage Gains to Decimal
For the given problem:
- First gain = 30% = 0.30
- Second gain = 20% = 0.20

### Step 3: Substitute the Gains into the Formula
We can express the final selling price (SP) in terms of the cost price (CP):
\[ \text{SP} = \text{CP} \times (1 + 0.30) \times (1 + 0.20) \]
\[ \text{SP} = \text{CP} \times (1.30) \times (1.20) \]

### Step 4: Calculate the Combined Gain Factor
Now, calculate the combined factor:
\[ 1.30 \times 1.20 = 1.56 \]
Thus, we can rewrite the equation for SP:
\[ \text{SP} = \text{CP} \times 1.56 \]

### Step 5: Set Up the Equation with Given Selling Price
We know the final selling price (SP) is Rs. 31,200:
\[ 31,200 = \text{CP} \times 1.56 \]

### Step 6: Solve for Cost Price (CP)
To find CP, rearrange the equation:
\[ \text{CP} = \frac{31,200}{1.56} \]

### Step 7: Perform the Division
Now, calculate the value:
\[ \text{CP} = \frac{31,200}{1.56} = 20,000 \]

### Final Answer
The cost price (CP) of the item is Rs. 20,000.

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