Problem 10 - Intermediate Algebra

Question

Cashews were mixed with \(p\) pounds of peanuts to make 100 pounds of a mixture. Write an algebraic expression that represents the number of pounds of cashews that were used. (PICTURE NOT COPY)

Step-by-Step Solution

Answer

The algebraic expression for the pounds of cashews is \( 100 - p \).

Step 1: Understanding the Scenario

You are given that cashews are mixed with peanuts to make a mixture that weighs a total of 100 pounds. We need to find out how many pounds of cashews are used in terms of peanuts.

Step 2: Determine the Expression for Cashews

Let the number of pounds of peanuts be represented by the variable \( p \). Since the total weight of the mixture is 100 pounds, the weight of the cashews can be represented as \( 100 - p \). This represents the number of pounds of cashews in the mixture.

Key Concepts

Mixture Problems

Mixture problems involve combining two or more substances to achieve a desired mix with specific properties such as weight, price, or concentration. These problems often appear in algebra and require understanding how these components interact. In the cashew and peanuts example, the goal is to find out the amount of one ingredient used, given the total weight of the mixture and the weight of the other ingredient. To solve these problems:

• Identify the total amount of the mixture.
• Determine known weights of some components.
• Use algebra to solve for the unknown component.

The core idea in a mixture problem is that the sum of the weights of all components equals the total weight of the mixture. This simple addition fact helps in forming equations that can be solved to find unknown quantities within the mixture.

Variable Representation

Variable representation is crucial when dealing with algebraic problems, especially mixture problems. A variable is a symbol that stands in for a number we want to find. In the example exercise, the variable used was "\( p \)" for representing the pounds of peanuts.When you use variables in algebra:

• They allow you to create equations representing relationships between different components.
• They provide a flexible way to handle numbers that are unknown or could change.
• They simplify complex problems into more manageable ones.

Careful representation of variables helps in clearly laying out the problem's structure. It ensures that each part of the equation reflects the real-world scenario it's meant to represent. In our example, once we know \( p \), the number of pounds of cashews can be easily computed as \( 100 - p \).

Mathematical Operations

Mathematical operations are fundamental to solving algebraic problems, including those involving mixtures. To understand and solve the cashew-peanut example, knowing how to perform basic operations like addition and subtraction is key.Key operations used include:

• Addition: Ensures that the combination of all parts equals the total. Here, the cashews and peanuts together total 100 pounds.
• Subtraction: Helps find unknown quantities by removing known amounts from the total. In this case, subtracting \( p \) pounds of peanuts from 100 pounds gives the weight of the cashews, \( 100 - p \).

These operations help manipulate equations to isolate variables and solve for unknowns. Mastering basic operations ensures a strong foundation in tackling not just algebraic expressions, but also a wide range of mathematical problems.