- Identify the proportionality constant \(k\) using the given data: \(k = \frac{\$4}{5 \text{ lb}} = \$0.80/\text{lb}\).
- Use the constant price to set up the linear equation \(y = 0.8x\).
- Substitute the desired weight, \(x = 10\) lb, into the equation to find the cost.
- The cost for 10 pounds of beans is \(\boxed{\$8}\).
Explanation
- Analyzing the Problem
To find the cost of 10 pounds of beans, we examine the relationship between the weight of the beans and their cost. We are given that 5 pounds of beans cost \(\$4\). Assuming a proportional relationship, the cost \(y\) in dollars for a given weight \(x\) in pounds can be represented by the equation \(y = kx\), where \(k\) is the constant cost per pound.
- Determining the Unit Price
First, we determine the unit price (the constant \(k\)) by using the provided data point of 5 pounds for \(\$4\).
Substituting \(x = 5\) and \(y = 4\) into the equation \(y = kx\), we get:
\[4 = k \cdot 5\]
To solve for \(k\), divide both sides by 5:
\[k = \frac{4}{5} = 0.8\]
Thus, the cost of beans is \(\$0.80\) per pound.
- Calculating the Cost for 10 Pounds
Now that we have the unit price \(k = 0.8\), we can calculate the cost for 10 pounds of beans. Substituting \(k = 0.8\) and \(x = 10\) into the proportional relationship \(y = kx\), we get:
\[y = 0.8 \cdot 10\]
\[y = 8\]
Therefore, 10 pounds of beans will cost \(\$8\).
Examples
Understanding proportional relationships is essential for everyday shopping. If you know the price of a certain amount of an item, you can easily calculate the cost for larger or smaller quantities. For instance, if you are buying fruit at a farmer's market, knowing that 2 pounds of apples cost \(\$3\) allows you to figure out the price for any weight, ensuring you can manage your budget effectively. This principle applies to everything from fuel costs to grocery shopping and even travel expenses.