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We are tasked with determining how much of each mixture needs to be used to produce 1000 kg of tropical juice. The tropical juice composition is 65% mango juice, 20% pineapple juice, and 15% water.

The three mixtures available are:
1. **Mixture (i)**: 62% mango juice, 12% pineapple juice, 26% water
2. **Mixture (ii)**: 92% pineapple juice, 8% water
3. **Mixture (iii)**: 98% mango juice, 2% water

Let:
- \( x_1 \) be the amount (in kg) of mixture (i) used.
- \( x_2 \) be the amount (in kg) of mixture (ii) used.
- \( x_3 \) be the amount (in kg) of mixture (iii) used.

We need to solve for \( x_1 \), \( x_2 \), and \( x_3 \) to satisfy the following constraints:
1. **Total weight constraint**:  
   \( x_1 + x_2 + x_3 = 1000 \) kg
2. **Mango juice content**:  
   The total amount of mango juice from all mixtures should equal 65% of 1000 kg, i.e., 650 kg.  
   So, \( 0.62x_1 + 0.98x_3 = 650 \)
3. **Pineapple juice content**:  
   The total amount of pineapple juice from all mixtures should equal 20% of 1000 kg, i.e., 200 kg.  
   So, \( 0.12x_1 + 0.92x_2 = 200 \)
4. **Water content**:  
   The total amount of water from all mixtures should equal 15% of 1000 kg, i.e., 150 kg.  
   So, \( 0.26x_1 + 0.08x_2 + 0.02x_3 = 150 \)

We now have a system of equations to solve:
1. \( x_1 + x_2 + x_3 = 1000 \)
2. \( 0.62x_1 + 0.98x_3 = 650 \)
3. \( 0.12x_1 + 0.92x_2 = 200 \)
4. \( 0.26x_1 + 0.08x_2 + 0.02x_3 = 150 \)

I'll solve this system of equations next.The solution to the system of equations gives the amounts of each mixture required to produce 1000 kg of tropical juice:

- Use approximately **503.75 kg** of mixture (i).
- Use approximately **151.69 kg** of mixture (ii).
- Use approximately **344.57 kg** of mixture (iii).

Would you like further details or explanations on any part of this solution?

Here are five related questions for further exploration:
1. How would the solution change if the target tropical juice had a different composition, e.g., 60% mango juice?
2. What if we only had two of the mixtures available? Could we still create the desired juice?
3. How would the equations change if we were producing 2000 kg of tropical juice instead of 1000 kg?
4. Can we derive a general formula for the amounts of each mixture for any tropical juice composition?
5. How can this system of equations be solved graphically?

**Tip**: In mixture problems, double-check your percentage compositions to ensure each component adds up correctly in your system of equations.

Tropical juice containing 65% (by weight) of mango juice, pineapple juice 20%, and 15% of water produced by adding the following juices: Mixture (i): 62% mango juice, 12% pineapple juice, and 26% water; Mixture (ii): 92% pineapple juice and 8% water; Mixture (iii): 98% mango juice and 2% water. How many kg of each mixture should be used to produce 1000 kg of tropical juice?

Learn how to solve a system of linear equations to determine how much of each mixture is needed to produce 1000 kg of tropical juice containing 65% mango juice, 20% pineapple juice, and 15% water. This problem covers mixture problems and the use of linear equation systems, suitable for Grades 11-12.

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