Mandy used the input and output in this table to write ratios. She concluded that because they are not all equivalent, this is not a proportional relationship. Is she correct? Explain.

Question

A 2-column table with 4 rows. Column 1 is labeled x with entries 1, 2, 5, 10. Column 2 is labeled y with entries 5, 10, 25, 50.

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Answer 1

To determine whether Mandy's conclusion is correct, let's first analyze the ratios formed by the input values (x) and the corresponding output values (y) from the table.

The table provides the following pairs of (x, y):

• For x = 1, y = 5
• For x = 2, y = 10
• For x = 5, y = 25
• For x = 10, y = 50

Now, we can calculate the ratios of y to x for each pair:

1. For (1, 5): \[ \frac{y}{x} = \frac{5}{1} = 5 \]

2. For (2, 10): \[ \frac{y}{x} = \frac{10}{2} = 5 \]

3. For (5, 25): \[ \frac{y}{x} = \frac{25}{5} = 5 \]

4. For (10, 50): \[ \frac{y}{x} = \frac{50}{10} = 5 \]

Now, we can see that all these ratios are equal to 5. Therefore, the ratios are:

\[ \frac{5}{1}, \frac{10}{2}, \frac{25}{5}, \frac{50}{10} \]

All these ratios simplify to 5, which indicates a consistent relationship between x and y.

Since all the ratios are equivalent and equal to 5, we can conclude that there is a proportional relationship between x and y.

Mandy's conclusion that the relationship is not proportional because the ratios are not equivalent is incorrect. In fact, the ratios are equivalent and consistently show a proportional relationship.