A shopkeeper has 12 kg of pure rice and 4 kg of impure rice. He mixes both type of rice but 4 kg of the mixture gets spoiled. If he agin mixes 4 kg of impure rice to the remaining mixture and professes to sell rice at a profit of `20%` on cost of pure rice. Then what is actual profit or loss percentage if cost of impure rice is `40%` of cost of pure rice.

Options

A. A)`(400)/(19)%`
B. B)`(250)/(21)%`
C. C)`(250)/(19) %`
D. D)`(500)/(19)%`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the cost price of pure and impure rice.
- Let the cost price (CP) of pure rice be \(10\) rupees per kg.
- Therefore, the cost price of impure rice, which is \(40\%\) of the cost of pure rice, will be:
\[ \text{Cost price of impure rice} = 0.4 \times 10 = 4 \text{ rupees per kg} \]

### Step 2: Calculate the total cost price of the rice mixture before spoilage.
- The shopkeeper has \(12\) kg of pure rice and \(4\) kg of impure rice.
- The total cost price of pure rice is:
\[ \text{CP of pure rice} = 12 \text{ kg} \times 10 \text{ rupees/kg} = 120 \text{ rupees} \]
- The total cost price of impure rice is:
\[ \text{CP of impure rice} = 4 \text{ kg} \times 4 \text{ rupees/kg} = 16 \text{ rupees} \]
- Therefore, the total cost price of the mixture before spoilage is:
\[ \text{Total CP} = 120 + 16 = 136 \text{ rupees} \]

### Step 3: Account for the spoilage of the mixture.
- \(4\) kg of the mixture gets spoiled. The remaining mixture will be:
\[ \text{Remaining mixture} = 12 \text{ kg} + 4 \text{ kg} - 4 \text{ kg} = 12 \text{ kg} \]

### Step 4: Mix additional impure rice.
- The shopkeeper adds another \(4\) kg of impure rice to the remaining mixture.
- The cost price of the additional \(4\) kg of impure rice is:
\[ \text{CP of additional impure rice} = 4 \text{ kg} \times 4 \text{ rupees/kg} = 16 \text{ rupees} \]
- The new total cost price after adding the impure rice is:
\[ \text{New Total CP} = 136 + 16 = 152 \text{ rupees} \]

### Step 5: Calculate the selling price.
- The selling price is based on a \(20\%\) profit on the cost of pure rice, which is \(10\) rupees per kg.
- The selling price per kg will be:
\[ \text{Selling Price per kg} = 10 + (20\% \text{ of } 10) = 10 + 2 = 12 \text{ rupees} \]
- The total selling price for \(16\) kg of the mixture is:
\[ \text{Total SP} = 12 \text{ rupees/kg} \times 16 \text{ kg} = 192 \text{ rupees} \]

### Step 6: Calculate profit and profit percentage.
- The profit made is:
\[ \text{Profit} = \text{Total SP} - \text{New Total CP} = 192 - 152 = 40 \text{ rupees} \]
- The profit percentage is calculated as:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{New Total CP}} \right) \times 100 = \left( \frac{40}{152} \right) \times 100 \]
- Simplifying this gives:
\[ \text{Profit Percentage} = \frac{4000}{152} \approx 26.32\% \]

### Final Answer:
The actual profit percentage is approximately \(26.32\%\).

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