- Calculate the ratio \(\frac{y}{x}\) for each given coordinate pair: \(\frac{30}{3}=10\), \(\frac{50}{5}=10\), and \(\frac{70}{7}=10\).
- Recognize that the constant of proportionality \(k\) is \(10\).
- Identify that Katerina's value of \(\frac{1}{10}\) is the inverse, \(\frac{x}{y}\), and she likely inverted the ratio \(\frac{y}{x}\).
- The final ratio is \(\boxed{10}\).
Explanation
- Analysis of the Proportional Relationship
A proportional relationship between two variables \(x\) and \(y\) is defined by the equation \(y = kx\), where \(k\) is the constant of proportionality. To find this constant, we can rearrange the equation to \(\frac{y}{x} = k\). We are given three pairs of \((x, y)\) values: \((3, 30)\), \((5, 50)\), and \((7, 70)\). Our goal is to determine the ratio \(\frac{y}{x}\) for each and evaluate Katerina's claim that \(\frac{y}{x} = \frac{1}{10}\).
- Calculating the Ratio \(\frac{y}{x}\)
We calculate the ratio \(\frac{y}{x}\) for each given pair:
- For \((3, 30)\): \(\frac{30}{3} = 10\).
- For \((5, 50)\): \(\frac{50}{5} = 10\).
- For \((7, 70)\): \(\frac{70}{7} = 10\).
The ratio \(\frac{y}{x}\) is consistently \(10\) for all provided values, meaning the constant of proportionality is \(k = 10\).
- Evaluating Katerina's Claim
Katerina claimed that the ratio \(\frac{y}{x} = \frac{1}{10}\). However, our calculations show that \(\frac{y}{x} = 10\). The value \(\frac{1}{10}\) is equal to \(\frac{x}{y}\) (e.g., \(\frac{3}{30} = \frac{1}{10}\), \(\frac{5}{50} = \frac{1}{10}\), and \(\frac{7}{70} = \frac{1}{10}\)). Therefore, Katerina likely inverted the fraction by calculating \(\frac{x}{y}\) instead of \(\frac{y}{x}\).
- Conclusion and Final Answer
The completed table is as follows:
| \(x\) | \(y\) | \(\frac{y}{x}\) |
| :--- | :--- | :--- |
| \(3\) | \(30\) | \(10\) |
| \(5\) | \(50\) | \(10\) |
| \(7\) | \(70\) | \(10\) |
Katerina's error was calculating \(\frac{x}{y}\) instead of \(\frac{y}{x}\). The correct ratio \(\frac{y}{x}\) is \(\boxed{10}\).
Examples
Understanding proportional relationships is essential in daily life, such as when calculating unit prices in a grocery store. If you buy 3 apples for \$30 (hypothetically), the price per apple is \$10. If you accidentally calculate the inverse, you would think each apple costs 1/10 of a dollar, which would lead to incorrect budgeting. Mastering these ratios ensures you accurately assess costs, speed, or any other relationship where one quantity increases at a constant rate relative to another.