Smartphone Profit Calculation: Finding Sujata's Purchase Price

This problem requires us to determine the original purchase price of a smartphone by reversing a series of transactions, each involving a specific profit percentage.

Profit Calculation Explained

The core concept here is profit calculation. Profit is the amount earned over the cost price (CP). When a profit percentage (P%) is applied, the selling price (SP) is calculated as:

$$ SP = CP \times \left(1 + \frac{P}{100}\right) $$

To find the original cost price when the selling price and profit percentage are known, we need to rearrange this formula:

$$ CP = \frac{SP}{\left(1 + \frac{P}{100}\right)} $$

We will use this inverse formula repeatedly to work backward from the final sale price to find the initial cost price.

Sequential Price Calculation

Let's denote the cost prices for Sujata, Anita, and Sameer as $CP_{Sujata}$, $CP_{Anita}$, and $CP_{Sameer}$ respectively. The final selling price, which is the price Ansh paid Sameer, is Rs. 45,540.

Sameer's Cost Price Calculation

Ansh bought the smartphone from Sameer for Rs. 45,540. This was Sameer's selling price ($SP_{Sameer}$), and it included a 10% profit for Sameer.

To find the price Sameer paid (his cost price, $CP_{Sameer}$), we apply the rearranged formula:

$$ CP_{Sameer} = \frac{SP_{Sameer}}{\left(1 + \frac{10}{100}\right)} $$

$$ CP_{Sameer} = \frac{45,540}{\left(1 + 0.10\right)} $$

$$ CP_{Sameer} = \frac{45,540}{1.10} $$

$$ CP_{Sameer} = 41,400 $$

This means Sameer purchased the phone for Rs. 41,400.

Anita's Cost Price Calculation

Sameer had purchased the phone from Anita. Therefore, Anita's selling price ($SP_{Anita}$) was Rs. 41,400 (Sameer's cost price). Anita had made a profit of 15% on this sale.

Now, we calculate Anita's cost price ($CP_{Anita}$):

$$ CP_{Anita} = \frac{SP_{Anita}}{\left(1 + \frac{15}{100}\right)} $$

$$ CP_{Anita} = \frac{41,400}{\left(1 + 0.15\right)} $$

$$ CP_{Anita} = \frac{41,400}{1.15} $$

$$ CP_{Anita} = 36,000 $$

Anita, therefore, bought the phone for Rs. 36,000.

Sujata's Cost Price Calculation

Anita had bought the phone from Sujata. This implies Sujata's selling price ($SP_{Sujata}$) was Rs. 36,000 (Anita's cost price). Sujata earned a profit of 20% from this sale.

The question asks for the price Sujata had originally bought the phone for, which is Sujata's cost price ($CP_{Sujata}$).

Calculating Sujata's cost price:

$$ CP_{Sujata} = \frac{SP_{Sujata}}{\left(1 + \frac{20}{100}\right)} $$

$$ CP_{Sujata} = \frac{36,000}{\left(1 + 0.20\right)} $$

$$ CP_{Sujata} = \frac{36,000}{1.20} $$

$$ CP_{Sujata} = 30,000 $$

Thus, Sujata originally purchased the smartphone for Rs. 30,000.

Transaction Summary Table

The sequence of transactions and calculated prices can be visualized as follows:

By applying the profit calculation formula in reverse, we successfully traced the price back to Sujata's original purchase cost.