Problem 78 - Beginning and Intermediate Algebra

Question

Write each phrase as an algebraic expression. Let \(x\) represent the unknown number. The quotient of a number and 9

Step-by-Step Solution

Answer

\(\frac{x}{9}\)

Step 1: Identify the Operation

The word 'quotient' indicates that you are dealing with division. Recall that a quotient is the result of dividing one number by another.

Step 2: Assign the Unknown Number

You are asked to express the phrase in terms of an unknown number, which we will denote as \(x\).

Step 3: Set Up the Division

The phrase states 'the quotient of a number and 9,' which translates algebraically to dividing our unknown number \(x\) by 9.

Step 4: Write the Algebraic Expression

Combine the identified operation (division) and the unknown \(x\) into an algebraic expression: \(\frac{x}{9}\).

Key Concepts

Understanding the Quotient

The term 'quotient' is integral in algebra and mathematics in general. It arises from division, where it's the result of dividing one number by another. For instance, in the division operation \(10 \div 2\), 5 is the quotient.

• A quotient tells us how many times a divisor can fit into a dividend.
• It helps compare values and solve equations.

Quotients are crucial in simplifying expressions and solving complex mathematical problems. It's important to recognize this in word problems, as it provides clues about the operations involved.

The Process of Division

Division is a basic arithmetic operation and is the method used to separate a total into equal parts. When we say 'the quotient of a number and 9', it indicates a division where you take the unknown number and divide it by 9.

• Division is represented by the symbol \(\div\) or, more commonly in algebra, as a fraction.
• It helps in determining how one quantity relates in size to another.

The importance of understanding division extends beyond basic arithmetic; it forms the foundation for solving algebraic equations and understanding fractions.

The Mystery of the Unknown Variable

In algebra, an unknown variable is commonly used to represent an unspecified number. It's the "mystery" part of algebraic equations that we often solve for. Here, the variable is \(x\), a placeholder for a number that we don't yet know.

• An unknown variable allows us to create general formulas and solve problems for any number.
• In expressions like \(\frac{x}{9}\), \(x\) could be any real number.

Variables make it possible to write expressions that can describe real-world relationships, giving algebra its power and utility in various fields. Recognizing and manipulating variables is a key skill in math.