- Calculate the constant of proportionality \(k_1 = \frac{y}{x}\) for Table 1, which yields \(k_1 = 2.5\).
- Calculate the constant of proportionality \(k_2 = \frac{y}{x}\) for Table 2, which yields \(k_2 = 1.4\).
- Compare the two constants: since \(2.5
eq 1.4\), the tables represent different proportional relationships.
- Conclusion: The two tables do not show the same proportional relationship, \(\boxed{\text{No}}\).
Explanation
- Analyzing the Concept of Proportionality
To determine if two tables represent the same proportional relationship, we must examine the ratio of the output variable \(y\) to the input variable \(x\) for each data set. A relationship is proportional if the ratio \(k = \frac{y}{x}\) remains constant for all data points within a table. Once we find the constant of proportionality \(k\) for each table, we can compare them to see if they describe the same mathematical relationship.
- Calculating the Constant of Proportionality for Table 1
For the first table, we calculate the ratio \(k_1 = \frac{y}{x}\) for each pair of data points:
- For \(x = 500, y = 1250\): \(\frac{1250}{500} = 2.5\)
- For \(x = 750, y = 1875\): \(\frac{1875}{750} = 2.5\)
- For \(x = 1000, y = 2500\): \(\frac{2500}{1000} = 2.5\)
Since the ratio is consistent across all pairs, the first table represents a proportional relationship defined by the equation \(y = 2.5x\).
- Calculating the Constant of Proportionality for Table 2
For the second table, we calculate the ratio \(k_2 = \frac{y}{x}\) for each pair:
- For \(x = 3, y = 4.2\): \(\frac{4.2}{3} = 1.4\)
- For \(x = 4, y = 5.6\): \(\frac{5.6}{4} = 1.4\)
- For \(x = 5, y = 7\): \(\frac{7}{5} = 1.4\)
Since the ratio is consistent, the second table also represents a proportional relationship, defined by the equation \(y = 1.4x\).
- Comparing the Relationships and Conclusion
We have determined that the first table corresponds to the relationship \(y = 2.5x\) and the second table corresponds to the relationship \(y = 1.4x\). Comparing the constants of proportionality, \(2.5
eq 1.4\), we conclude that the two tables do not show the same proportional relationship. Therefore, the answer is \(\boxed{\text{No}}\).
Examples
Proportional relationships appear in many everyday tasks, such as unit conversions or calculating costs. For example, if you are shopping, the price of apples might be proportional to their weight (e.g., \(\$1.40\) per pound). If you are preparing a recipe, the amount of flour required is proportional to the number of people being served. By calculating the constant of proportionality, you can easily predict the total cost or required ingredients for any quantity, ensuring your plans are accurate and efficient. Understanding these ratios helps you make informed decisions when balancing budgets or managing resources.