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Let's analyze the mathematical content from the image and provide the necessary explanations for each question:

1. **Question 1**: The problem involves calculating how many cans were thrown away given that 2.4 kg of aluminum is discarded. From the notes:
   - Each can weighs \( 0.16 \, \text{kg} \) (as noted).
   - To find the number of cans, you would divide the total weight by the weight of one can: 
     \[
     \frac{2.4 \, \text{kg}}{0.16 \, \text{kg}} = 15 \, \text{cans}
     \]
   So, 15 cans were thrown away.

2. **Question 2**: The recycled value is calculated for the cans.
   - If the weight of each can is \( 0.16 \, \text{kg} \), and there are 15 cans (as calculated in question 1), the total weight of the cans is \( 2.4 \, \text{kg} \).
   - The recycled value is provided in the notes as \( 0.25 \, \text{kg} \). Using the proportion for the 15 cans, we have:
     \[
     \frac{2.4 \, \text{kg}}{0.16 \, \text{kg}} \times 0.25 = 3.75
     \]
     Thus, the recycled value of those 15 cans is \( 3.75 \).

3. **Question 3**: This is about writing an equation to represent the number of cans (denoted as \( c \)) given their weight \( w \).
   - Since each can weighs \( 0.16 \, \text{kg} \), the equation can be written as:
     \[
     c = \frac{w}{0.16}
     \]
   This equation helps find the number of cans based on any given weight \( w \).

4. **Question 4**: This asks for an equation to represent the recycled value (denoted as \( v \)) of \( c \) cans.
   - If the recycled value of one can is \( 0.25 \), the equation becomes:
     \[
     v = 0.25 \times c
     \]
   Here, \( c \) is the number of cans.

5. **Question 5**: Write an equation to represent the recycled value of \( w \) kilograms of aluminum.
   - From earlier steps, we know each can weighs \( 0.16 \, \text{kg} \) and the recycled value per can is \( 0.25 \).
   - First, calculate the number of cans based on the weight \( w \):
     \[
     c = \frac{w}{0.16}
     \]
   - Then use this in the recycled value equation:
     \[
     v = 0.25 \times \frac{w}{0.16}
     \]
     Simplifying this:
     \[
     v = \frac{0.25}{0.16} \times w \approx 1.5625 \times w
     \]
   So, the equation is \( v = 1.5625 \times w \).

Would you like further details on any specific part? Here are five related questions to expand your understanding:

1. How would the equation change if the weight of each can was different?
2. What would the recycled value be for a total of 5 kg of aluminum?
3. How could you adjust these equations if the recycling price changes?
4. Can you calculate the total recycled value if you have 30 cans instead of 15?
5. How can this model be extended to a case where different types of cans with varying weights are recycled?

**Tip**: When writing equations based on real-world problems, always ensure you understand the relationships between variables like weight, quantity, and value before setting up the equation!

Calculate the number of aluminum cans, their recycled value, and write equations representing these relationships based on a given weight and value per can.

Learn how to calculate the number of aluminum cans and their recycled value based on their weight. This problem involves basic algebra and proportional relationships, suitable for students in Grades 6-8.

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