Problem 84 - Algebra for College Students

Question

Answer the question with an algebraic expression. The product of two numbers is 98 , and one of the numbers is \(n\). What is the other number?

Step-by-Step Solution

Answer

The other number is \( \frac{98}{n} \).

Step 1: Understand the problem

The problem gives us the product of two numbers as 98, with one number given as \( n \). We need to find the other number.

Step 2: Set up an equation

Since the product of two numbers is 98 and one of the numbers is \( n \), we can write the equation as \( n \times x = 98 \), where \( x \) is the number we need to find.

Step 3: Solve for x

To isolate \( x \), divide both sides of the equation \( n \times x = 98 \) by \( n \) to get \( x = \frac{98}{n} \).

Key Concepts

Solving Equations

Solving equations is all about finding the unknown value that satisfies a given mathematical statement. In the problem we're tackling, one equation is given: the product of two numbers equals 98 and one of those numbers is known, represented by the variable \( n \). Our task is to find the second number.

Setting up the equation from the problem statement, we have \( n \times x = 98 \). This is a nice straightforward equation for beginners to work with.

• Firstly, identify what is known and what needs to be found.
• Next, write down the equation that represents the problem using known algebraic rules.
• Finally, solve the equation to find the unknown by performing the necessary mathematical operations.

In this scenario, we isolated \( x \) to find the value that when multiplied by \( n \), gives 98.

Mathematical Problem-Solving

When tackling mathematical problems like this one, it is crucial to follow a structured approach which involves understanding, planning, and executing. In the product problem, we precisely use this method.

We begin with understanding the problem: identifying the knowns—one number \( n \) and their product 98—and unknowns—the other number.

With a clear understanding, the next step is to plan by forming an equation. This equation— \( n \times x = 98 \)—becomes our focus for action. By planning, you can visualize the steps required to isolate the variable \( x \). Execute by applying algebraic methods to solve the equation.

• The flow is simple: Understand → Plan → Execute.
• Organize your information logically so equation solving is seamless.
• Verify your result by substituting back to ensure it satisfies the original problem statement.

This disciplined problem-solving approach will serve you well in more complex math scenarios.

Variable Manipulation

Variable manipulation is a key skill in solving algebraic equations. It involves rearranging equations to isolate and determine the value of unknown variables. Here, we started with the equation \( n \times x = 98 \).

By dividing both sides by \( n \), we manipulate the equation to isolate \( x \), yielding \( x = \frac{98}{n} \).

This method of rearranging is crucial as it simplifies the problem and makes the variable's value clear.

• Remember that inverse operations help undo mathematical processes: multiplication is undone by division here.
• Pay attention to mathematical balance—whatever operation you perform on one side should be mirrored on the other.
• This ensures the equation's validity and leads to correct solutions.

Effective variable manipulation requires practice but becomes intuitive, allowing you to effortlessly dissect and solve different algebraic challenges.