How many kilograms of sugar at ₹3.80 per kg must be mixed with 60 kg of sugar at ₹4.60 per kg, so that by selling the mixture at ₹5.20 per kg there may be a gain of 30%?

Let's denote the amount of sugar to be mixed at \( \text{₹}3.80 \) per kg as \( x \) kg.

1. **Determine the total cost of each type of sugar:**
   - Cost of \( x \) kg of sugar at \( \text{₹}3.80 \) per kg: \( 3.80x \)
   - Cost of 60 kg of sugar at \( \text{₹}4.60 \) per kg: \( 60 \times 4.60 = 276 \)

2. **Determine the total quantity of the mixture:**
   - Total quantity = \( x + 60 \) kg

3. **Selling price and desired gain:**
   - Selling price per kg of mixture = \( \text{₹}5.20 \)
   - Gain of 30%

4. **Total selling price:**
   - Total selling price = \( (x + 60) \times 5.20 \)

5. **Cost price with the gain of 30%:**
   - Let the total cost price be \( \text{CP} \)
   - Selling price = \( \text{CP} \times (1 + \frac{30}{100}) = 1.3 \times \text{CP} \)

   Thus,
   \[
   (x + 60) \times 5.20 = 1.3 \times (3.80x + 276)
   \]

6. **Simplify the equation to find \( x \):**
   \[
   (x + 60) \times 5.20 = 1.3 \times (3.80x + 276)
   \]

   \[
   5.20x + 312 = 1.3 \times 3.80x + 1.3 \times 276
   \]

   \[
   5.20x + 312 = 4.94x + 358.8
   \]

   \[
   5.20x - 4.94x = 358.8 - 312
   \]

   \[
   0.26x = 46.8
   \]

   \[
   x = \frac{46.8}{0.26}
   \]

   \[
   x = 180
   \]

So, **180 kg** of sugar at \( \text{₹}3.80 \) per kg must be mixed with 60 kg of sugar at \( \text{₹}4.60 \) per kg to achieve the desired profit.

Would you like a detailed explanation of any specific step or have any questions?

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**Tip:** When dealing with mixture problems, always set up equations based on the total cost and the desired selling price to simplify finding the solution.

Learn how to calculate the amount of sugar at ₹3.80 per kg to mix with 60 kg at ₹4.60 per kg, ensuring a 30% gain when sold at ₹5.20 per kg. Detailed step-by-step solution provided.

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