Use the table to answer the question.

# of Trees # of Apples
2 26
3 39
6 78
10 130
12 156
Determine if the quantities in the data table are in a proportional relationship. If they are not in a proportional relationship, indicate why.

(1 point)
Responses

Yes, the data table has a proportional relationship.
Yes, the data table has a proportional relationship.

No, the ratios are not equivalent.
No, the ratios are not equivalent.

No, the ratio 6:78
is not proportional to the other ratios.
No, the ratio ratio of 6 to 78 is not proportional to the other ratios.

No, the table does not count consecutively.
No, the table does not count consecutively.

1 answer

To determine if the quantities in the data table are in a proportional relationship, we need to check if the ratio of the number of apples to the number of trees is constant for all entries in the table.

Let's calculate the ratios for each entry:

  1. For 2 trees: \( \frac{26 \text{ apples}}{2 \text{ trees}} = 13 \)
  2. For 3 trees: \( \frac{39 \text{ apples}}{3 \text{ trees}} = 13 \)
  3. For 6 trees: \( \frac{78 \text{ apples}}{6 \text{ trees}} = 13 \)
  4. For 10 trees: \( \frac{130 \text{ apples}}{10 \text{ trees}} = 13 \)
  5. For 12 trees: \( \frac{156 \text{ apples}}{12 \text{ trees}} = 13 \)

Since the ratio of apples to trees is consistently 13 across all entries, we can conclude that the data table does represent a proportional relationship.

Therefore, the correct response is:

Yes, the data table has a proportional relationship.