https://file.aiquickdraw.com/mathbot/file/1728575866433cmy19mlb.jpg

Let's address each of the questions in the provided image:

### 1. If a family threw away 2.4 kg of aluminum in a month, how many cans did they throw away? Explain or show your reasoning.
The weight of 10 aluminum cans is given as 0.16 kg. To find out how many cans were thrown away:

\[
\text{Number of cans} = \frac{2.4 \, \text{kg}}{0.16 \, \text{kg (per 10 cans)}} \times 10
\]

Simplifying:

\[
\text{Number of cans} = \frac{2.4}{0.16} = 15
\]

Thus, the family threw away 15 cans.

### 2. What would be the recycled value of those same cans? Explain or show your reasoning.
The value of 10 aluminum cans is given as $0.14. Since the family threw away 15 cans:

\[
\text{Recycled value} = \frac{15}{10} \times 0.14
\]

Calculating:

\[
\text{Recycled value} = 1.5 \times 0.14 = 0.21
\]

So, the recycled value of 15 cans would be $0.21.

### 3. Write an equation to represent the number of cans $ c $ given their weight $ w $.
The weight of 10 cans is 0.16 kg. Thus, the weight of one can is:

\[
\text{Weight of one can} = \frac{0.16}{10} = 0.016 \, \text{kg}
\]

The number of cans $ c $ given the weight $ w $ can be expressed as:

\[
c = \frac{w}{0.016}
\]

### 4. Write an equation to represent the recycled value $ r $ of $ c $ cans.
If 10 cans are worth $0.14, the value of one can is:

\[
\text{Value per can} = \frac{0.14}{10} = 0.014
\]

The recycled value $ r $ of $ c $ cans is:

\[
r = 0.014c
\]

### 5. Write an equation to represent the recycled value $ r $ of $ w $ kilograms of aluminum.
Using the relation $ c = \frac{w}{0.016} $ and $ r = 0.014c $, we substitute $ c $ into $ r $:

\[
r = 0.014 \times \frac{w}{0.016}
\]

Simplifying:

\[
r = 0.875w
\]

Thus, the recycled value $ r $ of $ w $ kilograms of aluminum is:

\[
r = 0.875w
\]

Let me know if you need any further clarification or details on these solutions.

---

### 5 Related Questions:
1. How would the number of cans change if the weight of 10 cans was increased to 0.2 kg?
2. If the value of 10 cans changed to $0.20, how would it affect the equations for recycled value?
3. What is the value of 50 cans using the derived formula?
4. If a family recycles 5 kg of aluminum, what would be the total value?
5. How many cans are needed to reach a recycled value of $1?

### Tip:
Always check units in equations to ensure consistency, particularly when dealing with weight and value conversions.

If a family threw away 2.4 kg of aluminum in a month, how many cans did they throw away? What would be the recycled value of those same cans?

Learn how to calculate the number of aluminum cans a family throws away based on weight and determine the recycled value using proportional relationships. This exercise is suitable for students in Grades 6-7 and covers basic algebraic and unit conversion concepts.

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