Problem 12 - Prealgebra

Question

Solve each problem by writing and solving an equation. Your friend bought 3 bags of wild birdseed and an \(\$ 18\) bird feeder. Each bag of birdseed costs the same amount. If your friend spent \$ 45,$ find the cost of one bag of birdseed.

Step-by-Step Solution

Answer

Each bag of birdseed costs \($9\).

Step 1: Understanding the Problem

We know that your friend bought a total of three bags of birdseed and a bird feeder, with the bird feeder costing \(\(18\). The total amount spent was \(\)45\). We are asked to find the cost of one bag of birdseed.

Step 2: Establish the Equation

Let's denote the cost of one bag of birdseed as \(x\). The total cost for the three bags is then \(3x\). Including the bird feeder, the total equation becomes \(3x + 18 = 45\).

Step 3: Solving the Equation

Now, we solve the equation \(3x + 18 = 45\) by first isolating the \(3x\) on one side. Subtract \(18\) from both sides: \(3x = 45 - 18\). This simplifies to \(3x = 27\).

Step 4: Finding the Cost of One Bag of Birdseed

Finally, solve for \(x\) by dividing both sides of the equation by \(3\): \(x = \frac{27}{3}\). Simplifying this gives \(x = 9\).

Step 5: Conclusion

Thus, the cost of one bag of birdseed is \($9\).

Key Concepts

Understanding Prealgebra

Prealgebra is the foundation of algebra, focusing on basic arithmetic and the introduction of algebraic concepts. It serves as a bridge between arithmetic and the more abstract ideas encountered in algebra. In this context, prealgebra allows us to transition from using numbers alone to incorporating variables like \(x\).
When dealing with problems in prealgebra, you'll often define an unknown value using a variable. This step is crucial. It sets the stage for solving equations. In our problem, we denote the cost of one bag of birdseed as \(x\). This approach transforms a word problem into a manageable equation, a skill that becomes increasingly important as you progress in mathematics.

Tackling Word Problems

Word problems can initially seem daunting, but they often follow a straightforward approach when broken down. The key is to translate the words into a mathematical equation.
To do this effectively:

• Read the problem thoroughly to understand what is being asked.
• Identify the known quantities and assign variables to unknowns.
• Write an equation that models the problem.

In the given example, we recognize a total purchase involving birdseed bags and a bird feeder. Understanding the roles of these components helps us construct an equation that captures the financial relationship described in the problem.
The journey from words to an equation allows us to apply logical reasoning to find solutions.

Applying Algebraic Reasoning

Algebraic reasoning involves thinking logically about the relationships between variables and numbers. It is a skill that enables us to systematically solve equations.
In our exercise, we write the equation \(3x + 18 = 45\). Here, we're acknowledging that the total cost can be expressed as a sum of the costs of the birdseed and the bird feeder.
The next step involves manipulating the equation to isolate the variable. By subtracting \(18\) from both sides, we focus solely on the birdseed expense.

• Subtract \(18\) to get: \(3x = 27\).
• Then, solve for \(x\) by dividing both sides by \(3\).

This process demonstrates how algebraic reasoning helps us figure out unknown quantities efficiently.

Mathematical Problem Solving

Mathematical problem solving involves using various strategies to find a solution to a problem. It is not just about getting the right answer but understanding the process and thinking involved.
In the example of finding the cost of a bag of birdseed, you're engaging in problem solving by:

• Identifying what you need to find.
• Setting up an equation based on the given information.
• Executing the steps to solve the equation.

The problem-solving process encourages critical thinking and the application of different mathematical concepts. By breaking the problem into smaller parts, we can efficiently work through each step. This approach not only helps in reach the correct solution but also builds a deeper understanding and confidence in handling similar problems.