Problem 73 - Algebra for College Students

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. The quotient of a number and 8

Step-by-Step Solution

Answer

The expression is \(\frac{n}{8}\).

Step 1: Identifying the Operation

The word 'quotient' indicates that we are dealing with division. Specifically, the phrase refers to dividing a number by another number.

Step 2: Assigning Variables

The phrase mentions 'a number,' which is the unknown quantity we need to represent with a variable. Let's use \(n\) for this unknown number.

Step 3: Writing the Algebraic Expression

Now, we need to express 'the quotient of a number and 8' in algebraic terms. Since the quotient involves division, the expression will be \(\frac{n}{8}\), representing the division of \(n\) by 8.

Key Concepts

Variables in Algebra

In algebra, variables are fundamental components that make it easier to express mathematical ideas. A variable is simply a symbol, often a letter, that stands for a number we don't know yet or that can change. In this exercise, we're using the letter \( n \) as a variable. This \( n \) stands for an unknown number that could take on different values.

• Variables help to generalize mathematical problems.
• They allow us to write expressions and equations that can be used for a range of numbers.
• Using \( n \) or any variable also helps in solving equations and understanding relationships between quantities.

When you see a variable in an algebraic expression, like \( n \) in our example, think of it as a placeholder for a variety of possible numbers. This flexibility is what makes algebra so useful for solving different types of problems.

Division in Algebra

Understanding division in algebra is key to correctly interpreting expressions that involve the word 'quotient.' A quotient is the result you get when you divide one number by another. When we talk about division in algebra, we are expressing how many times one quantity is contained within another.
For example, when the exercise refers to 'the quotient of a number and 8,' it's describing the division of that number by 8.

• This means we are breaking down the number \( n \) into parts of 8.
• The algebraic representation of this in our expression is \( \frac{n}{8} \).
• Division expressions show us how variables can be distributed or divided amongst other numbers.

When working with division in algebra, remember that you're interpreting how one number, the dividend, is separated into equal parts, as indicated by the divisor.

Writing Expressions in Algebra

Writing algebraic expressions is about translating verbal phrases into mathematical language. Each word or phrase in English can represent an operation or relationship between numbers in algebra. Here, we translate 'the quotient of a number and 8' into an expression.

• 'Quotient' suggests the operation of division, while 'a number' is represented by the variable \( n \).
• 'And 8' indicates that 8 is the divisor in this division relationship.
• The resulting expression \( \frac{n}{8} \) accurately represents the phrase using algebraic symbols.

Crafting these expressions requires carefully considering which operations correlate with which words, allowing us to convert spoken or written descriptions into clarity through formulas, like \( \frac{n}{8} \) in our exercise. This skill is useful for setting up equations based on real-life situations, solving for unknowns, and simplifying complex problems into comprehensible equations.