Problem 63 - Algebra 1

Question

To promote sales, a grocery store advertises bananas for \(\mathbf{S} . \mathbf{2 5}\) per pound. The store loses \(\mathbf{S} . \mathbf{1 1}\) on each pound of bananas it sells. The store also advertises apple juice for \(\$ 1.19\) per 64 -ounce bottle, and loses \(\$ .08\) per bottle sold. Use a verbal model to find how much the store loses on sales of 3107 bottles of apple juice.

Step-by-Step Solution

Answer

The store loses \$ 248.56 on sales of 3107 bottles of apple juice.

Step 1: Identify the loss per bottle of apple juice

The store makes a loss of \(\$ 0.08\) on each bottle of apple juice sold.

Step 2: Calculate total loss for apple juice

Calculate the total loss for all the apple juice sold by multiplying the loss per bottle (\$0.08) with the total number of bottles sold (3107). This is performed as follows: \(0.08 \times 3107\).

Step 3: Compute the result

Carry out the multiplication to find the total loss on the apple juice sales \(0.08 \times 3107 = \$ 248.56\).

Key Concepts

Understanding Loss Calculation

Calculating losses is a crucial skill in business and finance. It allows companies to assess how promotions or pricing strategies might impact their profits. In the exercise, the store experiences a loss of money on each product it sells. A loss calculation involves determining how much money a business loses based on multiple variables.

Factors involved in this process include:

• The cost of the item compared to its selling price
• The number of items sold
• Any additional costs related to the sale, like promotions

For the apple juice: the store loses \( \$0.08 \) per bottle. If the store sells multiple bottles, the total loss can be calculated by multiplying this loss per bottle by the number of bottles sold. This method helps businesses understand the financial impact of their selling prices and decide whether they need to adjust prices or promotions to prevent financial drawbacks.

The Role of Multiplication in Calculations

Multiplication plays a vital role in everyday calculations, especially when dealing with repetitive processes like sales. Instead of adding the loss for each individual bottle of apple juice, multiplication provides a faster and more efficient method.

In the context of the exercise, the multiplication concept is applied to determine the aggregate loss for a bulk amount of items. By calculating \( 0.08 \times 3107 \), we efficiently find the cumulative loss without the need for lengthy addition. This kind of calculation is foundational in subjects like algebra and calculus, where operations need to be computed quickly and accurately. By understanding multiplication's role, one realizes that it simplifies repetitious addition and, crucially, saves time in calculations involving financial data. This efficiency provides a streamlined approach that is invaluable not only in academics but also in real-world applications.

Solving Problems in Algebra Using Verbal Models

Algebra can seem intimidating, but using verbal models simplifies the process. A verbal model is a description that transforms a real-world scenario into an algebraic equation. It bridges the gap between a word problem and its numerical solution.

Take the example from the exercise: the loss for each bottle of apple juice (\(\$0.08\)) is a constant that applies to every item sold. By translating this idea into a simple multiplication formula, \[ \text{Total Loss} = \text{Loss per Item} \times \text{Number of Items Sold} \]we achieve clarity and comprehension. With the verbal model, even complex problems can be broken down into manageable parts, making problem-solving accessible to students and professionals alike. This approach sheds light on how variables interact within an equation and enhances the understanding of abstract algebraic concepts applied to tangible scenarios.