To determine whether the situation represents a proportional relationship, we need to analyze the relationship between the number of portraits painted and the time spent painting them.
First, we can calculate the rates of painting for the different scenarios:
-
2 portraits in 8 hours:
Rate = \( \frac{2 \text{ portraits}}{8 \text{ hours}} = \frac{1}{4} \text{ portraits per hour} \) -
3 portraits in 12 hours:
Rate = \( \frac{3 \text{ portraits}}{12 \text{ hours}} = \frac{1}{4} \text{ portraits per hour} \) -
5 portraits in 20 hours:
Rate = \( \frac{5 \text{ portraits}}{20 \text{ hours}} = \frac{1}{4} \text{ portraits per hour} \)
In this case, John paints at a constant rate of \( \frac{1}{4} \) portraits per hour, which indicates a linear relationship. However, a proportional relationship specifically requires that the ratio of the two quantities (portraits and time) is constant and that the relationship passes through the origin (0 portraits at 0 hours).
Now let's check if the relationship passes through the origin:
- If John spends 0 hours painting, he has painted 0 portraits.
This indeed indicates that the line passes through the origin.
Now, we must assess whether the relationship remains proportional. We check the ratio of portraits to hours for each scenario to confirm:
- For 2 portraits and 8 hours: \( \frac{2}{8} = \frac{1}{4} \)
- For 3 portraits and 12 hours: \( \frac{3}{12} = \frac{1}{4} \)
- For 5 portraits and 20 hours: \( \frac{5}{20} = \frac{1}{4} \)
Since the ratio is the same in all scenarios and it passes through the origin, this clearly indicates that it is a proportional relationship.
Therefore, the correct answer is:
B. The given situation represents a proportional relationship because the graph of the given data forms a straight line through the origin.