Question

John paints 2 portraits in 8 hours, 3 portraits in 12 hours, and 5 portraits in 20 hours. Which statement correctly describes this situation?
A.
The given situation does not represent a proportional relationship because the graph of the given data does not form a straight line through the origin.
B.
The given situation does not represent a proportional relationship because the graph of the given data forms a straight line through the origin.
C.
The given situation represents a proportional relationship because the graph of the given data forms a straight line through the origin.
D.
The given situation represents a proportional relationship because the graph of the given data does not form a straight line through the origin.

Answer 1

To determine whether the relationship is proportional, we need to check if the rate of portraits painted per hour is constant.

Let's calculate the rates for each case:

1. **For 2 portraits in 8 hours:**
\[
\text{Rate} = \frac{2 \text{ portraits}}{8 \text{ hours}} = \frac{1}{4} \text{ portraits per hour}
\]

2. **For 3 portraits in 12 hours:**
\[
\text{Rate} = \frac{3 \text{ portraits}}{12 \text{ hours}} = \frac{1}{4} \text{ portraits per hour}
\]

3. **For 5 portraits in 20 hours:**
\[
\text{Rate} = \frac{5 \text{ portraits}}{20 \text{ hours}} = \frac{1}{4} \text{ portraits per hour}
\]

In all cases, the rate is the same: \( \frac{1}{4} \) portraits per hour.

Since the rate of painting is consistent, this indicates a proportional relationship. In a proportional relationship, if the graph of the data is plotted, it should form a straight line through the origin (0,0).

Given these points:

- The relationship is proportional.
- The graph should form a straight line through the origin.

Thus, the correct statement is:

**C. The given situation represents a proportional relationship because the graph of the given data forms a straight line through the origin.**