Michael sold 2 flowers in 4 hours, 3 flowers in 6 hours, 4 flowers in 8 hours, and 5 flowers in 10 hours. Which statement correctly describes this situation?

Question

A.
The given situation does not represent a proportional relationship because the graph of the given data forms a straight line through the origin.
B.
The given situation represents a proportional relationship because the graph of the given data forms a straight line through the origin.
C.
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.
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 if the situation represents a proportional relationship, we can analyze the relationship between the number of flowers sold and the time taken.

Let's summarize the data:

• 2 flowers in 4 hours
• 3 flowers in 6 hours
• 4 flowers in 8 hours
• 5 flowers in 10 hours

In a proportional relationship, the ratio of the two quantities (flowers sold and time) remains constant, meaning that if you plot the points on a graph, the line must pass through the origin (0,0) and be straight (linear).

Now, calculating the rate (flowers per hour) for each case:

1. For 2 flowers in 4 hours: \( \frac{2 \text{ flowers}}{4 \text{ hours}} = 0.5 \text{ flowers/hour} \)
2. For 3 flowers in 6 hours: \( \frac{3 \text{ flowers}}{6 \text{ hours}} = 0.5 \text{ flowers/hour} \)
3. For 4 flowers in 8 hours: \( \frac{4 \text{ flowers}}{8 \text{ hours}} = 0.5 \text{ flowers/hour} \)
4. For 5 flowers in 10 hours: \( \frac{5 \text{ flowers}}{10 \text{ hours}} = 0.5 \text{ flowers/hour} \)

The rate remains constant at 0.5 flowers per hour, which is a consistent multiple of the hours.

Now, let's check if the points form a straight line through the origin:

• (4, 2), (6, 3), (8, 4), (10, 5) do not pass through the origin (0,0).

Since the graph of the data does not pass through the origin while maintaining a consistent ratio, the correct interpretation of the situation as it relates to proportionality is:

C. 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.

Answer 2

B

Answer 3

Let’s clarify the situation based on the data provided:

1. Points Given:

   • (4 hours, 2 flowers)
   • (6 hours, 3 flowers)
   • (8 hours, 4 flowers)
   • (10 hours, 5 flowers)

2. Checking for Proportionality:

   • For a relationship to be considered proportional, the quantities must maintain a constant ratio, and importantly, the graph of the relationship must pass through the origin (0,0).
   • Here, we can see that as time increases, the number of flowers sold also increases, but we need to check if the data points can be represented as \(y = kx\) where \(k\) is a constant.

3. Calculating the Ratios:

   • Rate: \( \frac{2 \text{ flowers}}{4 \text{ hours}} = 0.5\) (flowers/hour)
   • Rate: \( \frac{3 \text{ flowers}}{6 \text{ hours}} = 0.5\) (flowers/hour)
   • Rate: \( \frac{4 \text{ flowers}}{8 \text{ hours}} = 0.5\) (flowers/hour)
   • Rate: \( \frac{5 \text{ flowers}}{10 \text{ hours}} = 0.5\) (flowers/hour)

   Each of these calculations gives a consistent rate of 0.5 flowers per hour.

4. Intercept Check:

   • Although the rate or ratio is constant, when plotted, the points do not include the origin (0,0), indicating that it does not maintain a proportional relationship in the sense that proportional relationships must pass through the origin.

Therefore, the correct answer is C: 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.

Given your response indicating “B”, I understand you may have a different perspective. However, based on the definition of proportional relationships in mathematics, the key reasons above illustrate that option C accurately describes the situation.