Question

Write and solve an equation to answer the question. It takes 45 peanuts to make one ounce of peanut butter. How many peanuts will be needed to make a 12 -ounce jar of peanut butter?

Short answer

540 peanuts are needed to make a 12-ounce jar of peanut butter.

Step-by-step solution

Understand the problem

The problem tells the amount of peanuts needed to make 1 ounce of peanut butter. This can be represented as \(1 oz = 45\) peanuts. The problem asks how many peanuts are needed to make a 12 ounce jar of peanut butter.

Formulate an equation

Since the problem involves a direct relationship, the formula to use would be \(Amount\_in\_ounces \times Number\_of\_peanuts\_for\_1\_ounce = Total\_number\_of\_peanuts\). So plug the appropriate values into this formula to form an equation: \(12 \times 45 = x\), where \(x\) represents the total number of peanuts required.

Solve the equation

To solve for \(x\), simply do the multiplication: \(12 \times 45 = 540\).

Key concepts

Algebraic Equations

In mathematics, an algebraic equation is a statement of equality that involves variables and numerical constants. It's based on the principle that both sides of the equality symbol must have the same value. In the context of the given exercise, we see a simple algebraic equation being used to represent a real-world problem.

In the exercise, each ounce of peanut butter is equivalent to 45 peanuts which forms the equation \(1 \text{ oz} = 45 \text{ peanuts}\). When we want to know how many peanuts are needed for more ounces, we scale up the equation. The key to working with algebraic equations is to maintain balance. If we multiply one side by 12 (to represent the 12 ounces), we must do the same to the other side, resulting in the equation \(12 \text{ oz} \times 45 \text{ peanuts/oz} = x\), where \(x\) is the total number of peanuts needed.

Understanding the structure of algebraic equations is crucial in solving them effectively. Also, always be sure to identify and isolate the variable you are solving for; in this case, \(x\) represents our unknown value that we are looking to find.

Direct Variation

Direct variation is a fundamental concept in algebra where two quantities increase or decrease at the same rate. In the context of the problem, we have a direct relationship between the amount of peanut butter produced (in ounces) and the number of peanuts required.

This can be expressed with the equation \(y = k \times x\), where \(y\) is the total number of peanuts, \(x\) is the amount of peanut butter in ounces, and \(k\) is the constant of variation, describing how many peanuts are needed for each ounce. In our case, \(k = 45\), so our direct variation equation becomes \(y = 45 \times x\).

It's helpful to note that direct variation relationships can be visually represented by straight lines through the origin on a graph, with the slope being the constant of variation. In real-world scenarios, recognizing these relationships can significantly simplify the problem-solving process.

Problem-solving Strategies

Solving mathematical problems can sometimes be daunting, but with the right problem-solving strategies, we can tackle them more efficiently. The exercise provided is solved through a systematic approach. Here, the steps taken model an effective strategy for solving algebraic problems:

• Understand the problem: Recognize what the question is asking, which is how many peanuts are needed for a 12-ounce jar.
• Formulate an equation: Translate the given information into an algebraic equation. Recognizing the direct variation helps set up the equation \(x = 12 \times 45\).
• Solve the equation: Perform the necessary arithmetic (\(12 \times 45\)) to find the value of \(x\), which is the answer to the problem.

Applying a methodical approach allows for a clear path from the beginning to the end of a problem. Additionally, checking the solution is an important step not to overlook; ensure that the answer makes sense in the context of the problem. These strategies not only provide a solution but also build a strong foundation for solving more complex problems in the future.