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Let's break down the problem step by step to find Amal's cost price for syrup per kg:

### Given Data:
1. **Quantity of syrup:** 110 kg
2. **Quantity of juice:** 120 kg
3. **Syrup is 20% less costly than juice.** Let the cost price of juice per kg be \( J \). Then the cost price of syrup per kg will be \( 0.8J \).
4. **Profit on selling syrup (10 kg):** 10% profit
5. **Profit on selling juice (20 kg):** 20% profit
6. **Remaining mixture (after selling the first 30 kg):** Sold at ₹308.32 per kg with an overall profit of 64%.
7. **Goal:** Find the cost price of syrup.

### Step 1: Define Variables
- Let the cost price of juice be \( J \) per kg.
- The cost price of syrup is then \( 0.8J \) per kg.

### Step 2: Calculate Selling Prices for Initial Quantities
- **Selling price of 10 kg syrup at 10% profit:**
  \[
  \text{Selling price of syrup} = 10 \times 1.1 \times 0.8J = 8.8J
  \]
- **Selling price of 20 kg juice at 20% profit:**
  \[
  \text{Selling price of juice} = 20 \times 1.2 \times J = 24J
  \]

### Step 3: Remaining Mixture
After selling 10 kg of syrup and 20 kg of juice, the remaining quantities are:
- Remaining syrup: \( 110 - 10 = 100 \) kg
- Remaining juice: \( 120 - 20 = 100 \) kg

This remaining mixture (100 kg of syrup + 100 kg of juice = 200 kg) is sold at ₹308.32 per kg with a 64% profit.

- **Cost price of the remaining mixture per kg:**
  \[
  \text{Cost price of the mixture} = \frac{308.32}{1.64} = 188 \, \text{rupees per kg}
  \]

### Step 4: Total Cost of the Mixture
- The total cost price of the mixture is the weighted sum of syrup and juice costs:
  \[
  188 = \frac{100 \times 0.8J + 100 \times J}{200} = \frac{180J}{200} = 0.9J
  \]
Thus, 
\[
0.9J = 188 \quad \Rightarrow \quad J = \frac{188}{0.9} = 208.89 \, \text{rupees per kg}
\]

### Step 5: Cost Price of Syrup
The cost price of syrup is 20% less than that of juice:
\[
\text{Cost price of syrup} = 0.8 \times 208.89 = 167.11 \, \text{rupees per kg}
\]

### Final Answer:
Amal’s cost price for syrup is **₹167.11 per kg**.

Would you like more details, or do you have any questions?

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Here are 5 related questions for further practice:
1. How would the result change if Amal made a 50% profit on the mixture instead of 64%?
2. What would be the selling price if he sold 50 kg of the mixture at the same profit rate?
3. What is the total revenue from selling the entire quantity of syrup and juice?
4. If Amal's cost price for juice was reduced by 10%, how would that affect the final selling price?
5. What would be the cost price of syrup if juice was 25% more expensive instead of 20%?

### Tip:
When handling profit percentage problems, always ensure you convert profit percentages into multipliers (like 1.1 for 10% profit) to simplify calculations.

Amal buys 110 kg of syrup and 120 kg of juice, syrup being 20% less costly than juice, per kg. He sells 10 kg of syrup at 10% profit and 20 kg of juice at 20% profit. Mixing the remaining juice and syrup, Amal sells the mixture at ₹308.32 per kg and makes an overall profit of 64%. Then, Amal’s cost price for syrup, in rupees per kg, is?

Find out how to calculate the cost price of syrup when it's mixed with juice, sold at a profit, and the mixture's final price per kg is provided. This math problem covers key concepts such as percentage profit, weighted averages, and mixture problems.

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