Question

The table below displays the cost and the weight of five packages of chicken. Consider the weight as the x-value and the cost as the y-value and determine if a proportional relationship exists between cost and weight. Explain with a graph.



Solution

Here, the equivalent ratio is equation image indicator = equation image indicator = 6.99.

So, the equation would be y = 6.99x (since y = kx, where k is the factor of proportionality).

Let us draw a graph using the given table.



The graph is linear and passes through the origin.

The fact that one variable increases with the increase of the other variable is what makes the x and y values DIRECTLY proportional.

Answer 1

To determine if a proportional relationship exists between the cost and weight of the packages of chicken, we can analyze the relationship using the data provided in a hypothetical table and then graph it.

### Step 1: Establishing Proportionality

A relationship is proportional if the ratio of y (cost) to x (weight) remains constant for all pairs of values. Mathematically, this can be expressed as:

\[ \frac{y}{x} = k \]

where \( k \) is the constant of proportionality.

### Example Data

Let's say we have the following data for weight (in pounds) and cost (in dollars):

| Weight (x) | Cost (y) |
|------------|----------|
| 1 | 6.99 |
| 2 | 13.98 |
| 3 | 20.97 |
| 4 | 27.96 |
| 5 | 34.95 |

### Step 2: Finding the Ratio

Let's calculate the ratio \( \frac{y}{x} \) for each data point:

- For the first package: \( \frac{6.99}{1} = 6.99 \)
- For the second package: \( \frac{13.98}{2} = 6.99 \)
- For the third package: \( \frac{20.97}{3} = 6.99 \)
- For the fourth package: \( \frac{27.96}{4} = 6.99 \)
- For the fifth package: \( \frac{34.95}{5} = 6.99 \)

### Step 3: Conclusion About Proportionality

All values of \( \frac{y}{x} \) equal 6.99, indicating that the cost per pound of chicken remains constant. Therefore, there exists a proportional relationship between the cost and the weight of the packages.

### Step 4: Graphing the Relationship

To visualize this, plot the data points on a graph with 'Weight (x)' on the x-axis and 'Cost (y)' on the y-axis:

1. Plot the points (1, 6.99), (2, 13.98), (3, 20.97), (4, 27.96), and (5, 34.95).
2. Since all points lie on a straight line that passes through the origin (0, 0), this confirms a linear relationship.

### Step 5: Analyzing the Graph

- The graph is a straight line, indicating a linear relationship.
- It passes through the origin, which confirms that the relationship is proportional—when there is no weight (0 pounds), there is no cost (0 dollars).
- The slope of the line represents the constant \( k \), which in this case is 6.99.

### Conclusion

Since the graph is linear, passes through the origin, and the ratio of cost to weight remains constant at 6.99, we conclude that there is indeed a proportional relationship between the cost and the weight of the packages of chicken. Thus, the equation \( y = 6.99x \) accurately represents this relationship.