Problem 89 - Beginning and Intermediate Algebra

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. The quotient of 12 and a number is \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The equation is \(\frac{12}{x} = \frac{1}{2}\) and the solution is \(x = 24\).

Step 1: Understanding the Problem

The problem states: "The quotient of 12 and a number is \(\frac{1}{2}\)." We need to translate this sentence into a mathematical equation. The word "quotient" refers to division, and the unknown number can be represented as \(x\).

Step 2: Setting Up the Equation

Since the quotient of 12 and a number \(x\) is \(\frac{1}{2}\), we write this as an equation: \(\frac{12}{x} = \frac{1}{2}\).

Step 3: Solving the Equation

To solve the equation \(\frac{12}{x} = \frac{1}{2}\), we can use cross-multiplication to clear the fraction: \(12 \cdot 2 = 1 \cdot x\).

Step 4: Simplifying the Equation

Simplify the expression from cross-multiplication: \(24 = x\). This implies the value of the unknown number is 24.

Key Concepts

Understanding the Unknown Number in Algebra

In algebra, we often deal with problems where a specific number is not known. Such a number is referred to as the "unknown number." It is common practice to represent this unknown number using a variable—most frequently, the letter \( x \). For instance, in our example, we had the sentence: "The quotient of 12 and a number is \( \frac{1}{2} \)." Here, the unknown number is the value we are trying to identify, and it is denoted by \( x \). Our task is to figure out what number \( x \) represents to make the equation true.

Using a variable for the unknown number allows us to set up an equation based on the information given in the problem. This step is crucial because it provides a mathematical framework to work with variables systematically and solve it like a puzzle.

The Role and Method of Cross-Multiplication

Cross-multiplication is a technique used to solve equations that involve fractions. It is particularly handy when you have an equation in the form \( \frac{a}{b} = \frac{c}{d} \). To eliminate the fractions, cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction on each side of the equation.

Let's see how this works with our example: \( \frac{12}{x} = \frac{1}{2} \). By cross-multiplying, we multiply 12 by 2 and 1 by \( x \), resulting in:

• \( 12 \times 2 = 1 \times x \)

This simplifies to \( 24 = x \). So through cross-multiplication, we've effectively cleared the fractions and transformed the equation into a simple multiplication problem. This makes finding the unknown much easier.

Division Concepts in Algebraic Equations

Division is a fundamental operation in mathematics and plays a critical role in algebraic equations. When a problem mentions a "quotient," it signals that division is part of the equation. In our exercise, "The quotient of 12 and a number is \( \frac{1}{2} \)" referred to dividing 12 by an unknown \( x \).

In an algebraic context, division can often seem complex, especially when the divisor or dividend is an unknown.

• In the equation \( \frac{12}{x} = \frac{1}{2} \), the division of 12 by \( x \) needs to equal \( \frac{1}{2} \).
• By rearranging the equation or using techniques like cross-multiplication, we can resolve these complex divisions into simpler forms.

It's critical to understand that dividing by a variable doesn't change the fundamental rule of division but requires careful manipulation to isolate the unknown and solve it. With this understanding, division transitions from an abstract operation to a tool for solving practical problems.