To complete the table and analyze the error made by Katerina, let's first evaluate the proportional relationship between \( x \) and \( y \).
Given the values in the table:
- When \( x = 3 \), \( y = 30 \)
- When \( x = 5 \), \( y = 50 \)
- When \( x = 7 \), \( y = 70 \)
We can calculate the ratio \( \frac{y}{x} \) for each pair:
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For \( x = 3 \): \[ \frac{y}{x} = \frac{30}{3} = 10 \]
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For \( x = 5 \): \[ \frac{y}{x} = \frac{50}{5} = 10 \]
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For \( x = 7 \): \[ \frac{y}{x} = \frac{70}{7} = 10 \]
This indicates that the ratio \( \frac{y}{x} \) is consistently 10 for the values provided, not \( \frac{1}{10} \) as Katerina erroneously stated.
Now, we can fill in the table with the simplified ratios:
| \( x \) | \( y \) | Ratio \( \frac{y}{x} \) | |---------|---------|---------------------------| | 3 | 30 | 10 | | 5 | 50 | 10 | | 7 | 70 | 10 |
Likely Error
The error Katerina likely made was misunderstanding the proportional relationship and possibly miscalculating the division or misinterpreting the terms of the ratio. She stated the ratio as \( \frac{1}{10} \), which indicates a misunderstanding of how \( y \) is related to \( x \). The correct ratio demonstrates that \( y \) is actually 10 times \( x \), rather than \( x \) being one-tenth of \( y \).
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