Problem 48 - Algebra 1: Concepts and Skills

Question

MODELING REAL-LIFE PROBLEMS Use the verbal model to write a linear equation. Then use the multiplication property of equality to solve the equation. 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? \(\frac{\text { Number of peanuts }}{\text { Number of ounces }}=\) Number of peanuts per ounce

Step-by-Step Solution

Answer

The number of peanuts needed to make a 12-ounce jar of peanut butter is 540.

Step 1: Define the variables

Let's assign x to the number of peanuts and y to the number of ounces. So, we have 'x' peanuts and 'y' ounces.

Step 2: Verbal model conversion into mathematical equation

Based on the given information, we know that 45 peanuts equals 1 ounce, which gives us equation \(x/45 = y\).

Step 3: Substitute the given value and solve

We need to find out the number of peanuts would be needed for a 12 ounce jar of peanut butter. So, we substitute y with 12 into the equation, which gives us \(x/45 = 12\). Solving for x using multiplication property of equality, we get \(x = 45 * 12\).

Step 4: Solve for x

Finally, calculate the value for x, which will give us the number of peanuts needed for a 12 ounce jar of peanut butter. \(x = 45 * 12 = 540\)

Key Concepts

Real-life Problem Solving

Applying math to real-life problems allows us to understand how mathematics is present in our daily activities. In this scenario, we are trying to determine how many peanuts are necessary to make a 12-ounce jar of peanut butter.
Math helps us solve questions about quantity and measurements, like in cooking or budgeting.
In real-life problem solving, a verbal or word problem is often transformed into a mathematical equation to find the desired answer. To begin solving a real-life problem:

• Identify what you need to find out.
• Recognize known information in the problem.
• Systematically convert word statements into mathematical expressions.

In our example, we know the amount of peanuts per ounce and the size of the jar. Understanding how to transform this into a mathematical equation is vital in deriving the solution.

Variables in Algebra

Variables act as placeholders for unknown values or quantities. In algebra, they allow us to work with numbers in a general form, which is essential for solving equations.
In simpler terms, variables are like pronouns in language, standing in for things we don't know yet. When dealing with linear equations, variables allow us to understand and manipulate the relationship between different quantities. In our example, we use 'x' to represent the number of peanuts needed and 'y' to denote the ounces of peanut butter.

• This helps us set up the relationship: number of peanuts per ounce.
• By using variables, we can create equations that model the scene in a meaningful way.
• Variables enable us to plug in different numbers to solve similar problems in various scenarios.

By defining variables clearly, we can ensure the solution follows logically from the initial problem setup.

Multiplication Property of Equality

The multiplication property of equality is a fundamental principle that helps us solve equations. This property states that if you multiply both sides of an equation by the same non-zero number, the sides remain equal.
This is useful in maintaining balance on both sides of an equation when you're trying to isolate a variable.In our peanuts and peanut butter problem, we used this property to solve for 'x', the number of peanuts, after substituting 'y' with 12:

• Starting with the equation \(x/45 = 12\).
• We multiply both sides by 45 to get \(x = 45 \times 12\).
• Calculating gives us \(x = 540\), showing us the number of peanuts needed.

This property is part of what makes solving equations systematic and reliable. Knowing when and how to use it can unlock solutions to many algebraic problems.