Problem 53 - Algebra A Combined Function

Question

Solve. Suppose you mix an amount of candy costing \(\$ 0.49\) a pound with candy costing \(\$ 0.65\) a pound. Which of the following costs per pound could result? a. \(\$ 0.58\) b. \(\$ 0.72\) c. \(\$ 0.29\)

Step-by-Step Solution

Answer

Only option a (\$0.58) is a possible cost.

Step 1: Identify Lowest and Highest Possible Costs

The lowest cost per pound for any mix would be \\(0.49\ (if only the cheaper candy is used), and the highest cost per pound would be \\)0.65\ (if only the more expensive candy is used). So, the resulting cost per pound must be between \\(0.49\ and \\)0.65\ inclusive.

Step 2: Analyze Option a (\$0.58)

Check if \\(0.58\ is within the range [\\)0.49, \\(0.65]. Since \\)0.58\ is greater than \\(0.49\ and less than \\)0.65\, it is a possible cost per pound.

Step 3: Analyze Option b (\$0.72)

Check if \\(0.72\ is within the range [\\)0.49, \\(0.65]. Since \\)0.72\ is greater than \$0.65\, it cannot be a possible cost per pound.

Step 4: Analyze Option c (\$0.29)

Check if \\(0.29\ is within the range [\\)0.49, \\(0.65]. Since \\)0.29\ is less than \$0.49\, it cannot be a possible cost per pound.

Key Concepts

Cost Analysis

Understanding cost analysis in mixture problems can help you determine the range of possible prices when combining two different items. Imagine you have two types of candy, one priced at \\(0.49 and the other at \\)0.65 per pound. The goal is to figure out the cost of a blend of these two candies.

The price per pound of the mixture depends on the proportion of each type of candy you include. If you only use the \\(0.49 candy, it makes sense that your blend would cost \\)0.49 per pound. Conversely, if you only use the \\(0.65 candy, then it would cost \\)0.65 per pound. Therefore, any mixture must have a cost that falls between these two prices.

An effective approach is calculating the boundary prices, which means the range from the lowest cost to the highest cost of the candies involved. For this problem, that boundary is \[0.49, 0.65\]. Any possible mixture price must fit within this range.

Algebraic Reasoning

In mixture problems, algebraic reasoning helps us determine which combinations are possible. Using algebra, we can express the problem in equations that represent real-world mixtures. This involves understanding how different weights, values, or volumes contribute to the final outcome.

Consider the candy example:

• You have candy \(A\) at \\(0.49/pound and candy \(B\) at \\)0.65/pound.
• Let \(x\) be the number of pounds of candy \(A\) and \(y\) be the number of pounds of candy \(B\).
• To find the cost per pound, you would form an equation based on their costs and weights: \(\text{Total cost} = 0.49x + 0.65y\).
• To form an algebraic expression for the price per pound of the mixture, you divide the total cost by \((x + y)\).

Solving such equations will let you test whether a specific price is logically feasible in relation to the proportions of the products involved.

Price Boundaries

Price boundaries are essential in mixture problems as they set the limits for what can be considered a feasible price for a blended product. The reason they matter is because they confirm the realism of calculated prices in terms of the components used.

The blend of candies in the exercise gives us a clear example:

• Lowest boundary: determined if your mixture is entirely made of the cheaper candy, here it's \\(0.49.
• Highest boundary: happens if the mixture uses only the expensive candy, so \\)0.65 in this case.
• Your feasible prices, therefore, lie somewhere between these two; in this example, the range [0.49, 0.65].

To determine if a given possibility like \\(0.58, \\)0.72, or \\(0.29 is valid, just check if it falls within these boundaries.

• \\)0.58 does, meaning it's possible.
• Neither \\(0.72 nor \\)0.29 fit within these boundaries, thus making them impossible outcomes.

Understanding price boundaries helps you quickly assess the validity of potential mixture costs.