Sunil buys 2500 Kg of weat, 1/5 of which he sells at a gain of 5%, 1/4 at a gain of 10%, 1/ 2 at a gain of 12% and the remainder at a gain of 16%. If he had sold the whole wheat at a gain of 11%. He would have made 7 87.50 more. What was the cost price of wheat per Kg.

Options

A. a)Rs 4
B. b)Rs 5
C. c)Rs 6
D. d)Rs 7

Answer

To solve the problem step by step, let's break down the calculations clearly.

### Step 1: Define the Cost Price
Let the cost price (CP) of wheat per kg be \( x \) rupees.

### Step 2: Calculate Total Cost Price
Since Sunil buys 2500 kg of wheat, the total cost price will be:
\[ \text{Total CP} = 2500 \times x = 2500x \text{ rupees} \]

### Step 3: Calculate Gains from Each Portion of Wheat Sold
1. **First Portion**: \( \frac{1}{5} \) of 2500 kg is sold at a gain of 5%.
\[ \text{Weight sold} = \frac{1}{5} \times 2500 = 500 \text{ kg} \]
\[ \text{Gain} = 500 \times x \times \frac{5}{100} = 25x \text{ rupees} \]

2. **Second Portion**: \( \frac{1}{4} \) of 2500 kg is sold at a gain of 10%.
\[ \text{Weight sold} = \frac{1}{4} \times 2500 = 625 \text{ kg} \]
\[ \text{Gain} = 625 \times x \times \frac{10}{100} = 62.5x \text{ rupees} \]

3. **Third Portion**: \( \frac{1}{2} \) of 2500 kg is sold at a gain of 12%.
\[ \text{Weight sold} = \frac{1}{2} \times 2500 = 1250 \text{ kg} \]
\[ \text{Gain} = 1250 \times x \times \frac{12}{100} = 150x \text{ rupees} \]

4. **Remaining Portion**: The remainder is sold at a gain of 16%.
\[ \text{Weight sold} = 2500 - (500 + 625 + 1250) = 1125 \text{ kg} \]
\[ \text{Gain} = 1125 \times x \times \frac{16}{100} = 180x \text{ rupees} \]

### Step 4: Calculate Total Gain
Now, we sum all the gains:
\[ \text{Total Gain} = 25x + 62.5x + 150x + 180x = 417.5x \text{ rupees} \]

### Step 5: Calculate Gain if Sold at 11%
If Sunil sold the whole wheat at a gain of 11%, the gain would be:
\[ \text{Gain at 11%} = 2500 \times x \times \frac{11}{100} = 275x \text{ rupees} \]

### Step 6: Set Up the Equation
According to the problem, if he sold the whole wheat at a gain of 11%, he would have made 87.50 rupees more than the total gain from selling in portions:
\[ 275x = 417.5x + 87.5 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 275x - 417.5x = 87.5 \]
\[ -142.5x = 87.5 \]
\[ 142.5x = -87.5 \]

### Step 8: Solve for x
Now, divide both sides by -142.5:
\[ x = \frac{87.5}{142.5} \]
To simplify:
\[ x = \frac{875}{1425} = \frac{5}{8} = 5 \text{ rupees} \]

### Conclusion
Thus, the cost price of wheat per kg is:
\[ \boxed{5 \text{ rupees}} \]