Problem 49 - Algebra 1

Question

The science club is selling magazine subscriptions at \(\$ 15\) each. The club wants to raise \(\$ 315\) for science equipment. Write a verbal model that relates the number of subscriptions, the cost of each subscription, and the amount of money the club needs to raise.

Step-by-Step Solution

Answer

The science club needs to sell 21 subscriptions to raise \$315.

Step 1: Identify the elements

Identify all the key elements from the problem. The number of subscriptions is unknown, therefore it can be represented by \(x\). The cost of each subscription is \$15 and the total amount the club wants to raise is \$315.

Step 2: Form the equation

Form an equation based on the elements of the problem. This represents a simple linear equation. So, by applying the formula 'number of items times cost of each item equals total cost', the equation should be \(15x = 315\).

Step 3: Solve the equation

After forming the equation, solve it to find the unknown variable \(x\). To get the value of \(x\), divide both sides of the equation by 15. Thus, the equation becomes \(x = 315 / 15\).

Step 4: Find the number of subscriptions

Perform the division operation to find the number of subscriptions. \(x = 315 / 15\) becomes \(x = 21\)

Key Concepts

Verbal Model

When tackling word problems, a verbal model is an essential tool that helps you to translate the situation into an equation. To create a verbal model, you should first recognize the question being asked and then determine the components that can influence the answer.

For the magazine subscription scenario, the verbal model should establish a relationship between three key components: the number of subscriptions sold (which we don't yet know), the price per subscription, and the total amount of money needed for the science equipment. The verbal formula could be ‘number of subscriptions times the price per subscription equals the total amount needed'. This verbal model serves as a framework for crafting an algebraic expression that will solve the problem.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (such as addition and multiplication). In word problems, setting up the correct algebraic expression is critical.

In our magazine subscription problem, we express the unknown number of subscriptions as x. Since each subscription is sold at \(15, we can write the algebraic expression as 15x. This represents the total revenue from selling x subscriptions at \)15 each and is a crucial step in developing the equation needed to find the solution.

Equation Solving

Equation solving is the process of finding the value of variables that satisfy a given mathematical equation. Once the algebraic expression is set up, creating and solving the equation is the next step. For linear equations with one variable, like our example, the solution often involves basic operations: addition, subtraction, multiplication, or division.

The equation formed from the problem, 15x = 315, requires us to isolate the variable x. This is achieved by dividing both sides of the equation by 15, resulting in x = 21. By solving this equation, we understand that to reach their fundraising goal, the science club needs to sell 21 magazine subscriptions.

Word Problem

Word problems are literary descriptions of mathematical scenarios that require interpretation and manipulation to uncover numerical solutions. They range from simple to complex and often incorporate real-life situations.

To effectively solve a word problem like the one posed by the science club's fundraiser, it's important to carefully read the problem, identify key information and variables, establish a verbal model, translate it into an algebraic expression, and finally solve the resulting equation. As these problems represent real-world issues, solving them enhances critical thinking and practical mathematical application skills.

Incorporating the exercise improvement advice requires us to ensure clear identification of each component and step involved in solving the word problem, such as outlining the verbal model, crafting the algebraic expression, and methodically solving the equation.