Find the Error - One cleaning solutions uses 1 part vinegar with 2 parts water. Another cleaning soluation uses 2 parts vinegar with 3 parts water. A student says that this represents a proportional relationship because in each soluation, there is one more part of water than vinegar. Find the error and correct it.

The student's reasoning is incorrect because they are misunderstanding what it means for two ratios to be proportional. In mathematics, two ratios are proportional if they represent the same fraction or relationship when simplified.

Let's examine the two cleaning solutions:

1. **First solution**: 1 part vinegar to 2 parts water.  
   - Ratio: \(\frac{1 \text{ part vinegar}}{2 \text{ parts water}} = \frac{1}{2}\)

2. **Second solution**: 2 parts vinegar to 3 parts water.  
   - Ratio: \(\frac{2 \text{ parts vinegar}}{3 \text{ parts water}} = \frac{2}{3}\)

These two ratios, \(\frac{1}{2}\) and \(\frac{2}{3}\), are not equal. Therefore, they do not represent a proportional relationship. 

The student made an error by focusing on the difference between the parts of water and vinegar (which is 1 in both cases) rather than the actual ratios of vinegar to water. The correct way to determine if they are proportional is by checking if the ratios themselves are equivalent, which they are not.

### To Correct the Student's Understanding:

The proportional relationship is not determined by the difference between parts of water and vinegar but by the ratio of vinegar to water. Since \(\frac{1}{2}\) is not equal to \(\frac{2}{3}\), the two solutions do not represent a proportional relationship.

Would you like further details on ratios and proportions or have any other questions?

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Here are five related questions for further exploration:
1. What does it mean for two ratios to be proportional?
2. How can you determine if two ratios are proportional?
3. What are other examples of proportional relationships in real life?
4. How would you solve for an unknown quantity in a proportion?
5. How can proportional reasoning be applied to solve word problems?

**Tip:** When comparing ratios, always convert them to fractions or decimals to easily determine if they are equivalent.

Explore how to solve a problem involving cleaning solutions and ratios. Learn about ratios and proportions in mathematics with a detailed explanation and examples. Suitable for Grades 6-8 students.

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