- Determine the cost per pound by calculating the ratio \(\frac{y}{x}\) for the given data points, which is \(\frac{4}{5} = 0.8\) dollars per pound.
- Confirm the relationship is proportional since the ratio is constant.
- Use the unit rate to find the cost for \(15\) pounds by calculating \(0.8 \times 15\).
- Conclude that the cost for \(15\) pounds of beans is \(\boxed{\$12}\).
Explanation
- Analyzing the Data and Identifying the Relationship
We are given a table showing the relationship between the weight of beans in pounds (\(x\)) and the corresponding cost in dollars (\(y\)). The data points provided are:
- When \(x = 5\) lb, the cost is \(y = \$4\).
- When \(x = 10\) lb, the cost is \(y = \$8\).
To find the cost for \(15\) pounds of beans, we first need to determine if the relationship between the weight and the cost is proportional, which means the cost per pound remains constant. We calculate the unit rate \(k = \frac{y}{x}\) for the given points:
- For \(x = 5, y = 4\): \(k = \frac{4}{5} = 0.8\) dollars per pound.
- For \(x = 10, y = 8\): \(k = \frac{8}{10} = 0.8\) dollars per pound.
Since the unit rate is constant, the cost follows the proportional relationship \(y = 0.8x\).
- Calculating the Cost for 15 Pounds
Now that we have established the unit rate of \(\$0.8\) per pound, we can determine the cost for \(15\) pounds of beans by substituting \(x = 15\) into our linear equation:
\[ y = 0.8 \times 15 \]
Performing the multiplication:
\[ y = 12 \]
Thus, the cost for \(15\) pounds of beans is \(\$12\).
Examples
Understanding proportional relationships is essential for everyday tasks, like shopping. For example, if you know the price of a certain amount of fruit, you can calculate the price of any other amount using the same unit rate. This helps you manage your budget and make smart purchasing decisions in grocery stores or at markets.