Problem 85 - Algebra for College Students

Question

Answer the question with an algebraic expression. The quotient of two numbers is 8 , and the smaller number is \(y\). What is the other number?

Step-by-Step Solution

Answer

The other number is \(8y\).

Step 1: Understanding the Problem

We are told that the quotient of two numbers is 8, and one of the numbers is represented by the variable \(y\). We need to find the other number by expressing it algebraically.

Step 2: Set Up the Equation

The quotient of two numbers means we are dividing one number by another. In this case, we can write the equation as: \[ \frac{x}{y} = 8 \] where \(x\) is the larger number we are trying to find.

Step 3: Solve for the Unknown

To find \(x\), we solve the equation by manipulating it algebraically. Start by multiplying both sides of the equation by \(y\) to isolate \(x\): \[ x = 8y \] This equation tells us that the larger number is 8 times the smaller number \(y\).

Key Concepts

Understanding Quotients

In algebra, a quotient is the result of dividing one number by another. For example, if you see a statement like "the quotient of 8 and 2 is 4," it means that when 8 is divided by 2, the result is 4. Quotients are often represented using the division symbol, as in \( \frac{x}{y} \). To understand a quotient, you should keep in mind:

• The numerator is the number being divided.
• The denominator is the number you are dividing by.
• The result of this division is the quotient.

In the given problem, the quotient of the two numbers is stated as 8, indicating how many times the smaller number \(y\) fits into the larger number. Recognizing this relationship is crucial in setting up the equation properly.

Formulating Equations

An equation is a mathematical statement that asserts the equality of two expressions. When you solve algebraic problems, such as the one given, you're often creating an equation as a way to find an unknown value. In our problem:

• We have two numbers where their quotient is 8.
• One of the numbers is \(y\), and the other is \(x\).

This leads us to the equation \( \frac{x}{y} = 8 \), which signifies that dividing \(x\) by \(y\) gives us 8. The goal is to solve this equation to find the value of the unknown, \(x\). Creating such equations involves:

• Understanding the relationship between the numbers as stated in the problem.
• Translating that relationship into a mathematical representation.

Through this process, equations become a powerful tool to analyze and solve real-world problems.

Techniques of Variable Isolation

Isolating a variable is a key step in solving algebraic equations. It means getting the unknown variable alone on one side of the equation, which helps in finding its value. In our example, we want to solve for \(x\) in the equation \( \frac{x}{y} = 8 \). To achieve variable isolation, follow these steps:

• Multiply both sides of the equation by \(y\) to cancel out the denominator and isolate \(x\) on one side: \(x = 8y\).
• This step is crucial because it transforms the equation into a simpler form that directly shows the relationship between \(x\) and \(y\).

By isolating \(x\), we discover that the other number is 8 times the smaller number \(y\). This process of rearranging and solving the equation to isolate the variable is fundamental to algebra and helps in solving many similar problems.