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This worksheet includes various problems centered around understanding and solving proportional relationships. Here's a breakdown of how you can approach each question:

1. **Identify Proportional Relationships in Tables**: Check if the ratio between corresponding values in each table remains constant to identify proportionality.

2. **Proportional Relationships in Graphs**: Proportional relationships are represented by straight lines that pass through the origin. For these, the equation takes the form \( y = kx \), where \( k \) is the constant of proportionality.

3. **Characteristics of Proportional Graphs**: Generally, these graphs are straight lines through the origin, and the equation is \( y = kx \), with \( k \) as the constant.

4. **Situational Analysis of Proportionality**:
   - (a) and (c) might show proportional relationships depending on if there is a fixed charge involved or not.
   - Situations with additional fixed fees (like in (b)) are not proportional.

5-15. **Problem Solving with Proportions**:
   - **Problem 5**: Use the unit rate of tents produced per minute to determine the time required.
   - **Problem 6-10**: Solve using direct proportionality (e.g., \( \frac{12 \text{ cans}}{4 \text{ days}} = \frac{x \text{ cans}}{2 \text{ days}} \)).
   - **Problem 11**: Compare lap rates to determine who is faster.
   - **Problem 12-13**: For medication and dilution, calculate proportional quantities using the given data.
   - **Problem 14**: Analyze area and perimeter relationships in the square design series.
   - **Problem 15**: Combine machine production rates to solve for total output within specified time constraints.

Would you like detailed solutions to any specific question or further clarification on any of these concepts?

### Further Questions:
1. How can we determine if a relationship is proportional without a graph?
2. What are common real-life examples of proportional relationships?
3. How does a fixed charge affect proportionality?
4. What is the significance of the constant \( k \) in proportional equations?
5. How can unit rates simplify solving proportional problems?

### Tip:
When identifying proportionality in a table or situation, look for a constant multiplier between pairs of values or a constant rate in equations.

This worksheet includes various problems centered around understanding and solving proportional relationships. Here's a breakdown of how you can approach each question.

This worksheet covers various problems involving proportional relationships, ratios, and solving equations. It provides a clear breakdown of proportionality in tables, graphs, and real-life scenarios. Learn how to solve problems using key formulas like y = kx. Suitable for Grades 7-9.

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