Teresa makes homemade empanadas and sells them at local events. The table shows the price she charges for different quantities of empanadas. Is the relationship between price and number of empanadas proportional? Complete the statements.
The price per empanada is the same for each quantity purchased, so the relationship is proportional.
| Empanadas | Price (dollars) |
|---|---|
| 2 | 1 |
| 4 | 2 |
| 6 | 3 |
| 8 | 4 |
The constant of proportionality is $\square$.
Answer (2)
Calculate the price per empanada for each quantity: 2 1 = 0.50 , 4 2 = 0.50 , 6 3 = 0.50 , 8 4 = 0.50 .
Observe that the price per empanada is the same for all quantities.
Conclude that the relationship is proportional.
State the constant of proportionality: 0.50 .
Explanation
Analyzing the Problem Let's analyze the given data to determine if the relationship between the number of empanadas and the price is proportional. A relationship is proportional if the ratio between two quantities is constant. In this case, we need to check if the price per empanada is the same regardless of the quantity purchased.
Checking for Proportionality To check for proportionality, we will calculate the price per empanada for each given quantity:
For 2 empanadas: The price is 1 , so t h e p r i ce p ere m p ana d ai s \frac{1}{2} = $0.50. For 4 empanadas: The price is 2 , so t h e p r i ce p ere m p ana d ai s \frac{2}{4} = $0.50. For 6 empanadas: The price is 3 , so t h e p r i ce p ere m p ana d ai s \frac{3}{6} = $0.50. For 8 empanadas: The price is 4 , so t h e p r i ce p ere m p ana d ai s \frac{4}{8} = $0.50.
Determining the Constant of Proportionality Since the price per empanada is $0.50 for all given quantities, the relationship between the number of empanadas and the price is proportional. The constant of proportionality is the price per empanada, which is $0.50.
Final Answer Therefore, the relationship is proportional, and the constant of proportionality is 0.50.
Examples
Understanding proportional relationships is useful in everyday scenarios such as calculating the cost of items when buying in bulk, determining the amount of ingredients needed when scaling up a recipe, or converting between different units of measurement. For instance, if you know that 2 empanadas cost $1, you can use the constant of proportionality ($0.50 per empanada) to quickly calculate the cost of any number of empanadas. If you wanted to buy 10 empanadas, you would simply multiply 10 by $0.50 to find that it would cost you $5.
The relationship between the price and the number of empanadas is proportional because the price per empanada remains constant at $0.50 across varying quantities. Therefore, the constant of proportionality is 0.50.
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