Problem 73 - Intermediate Algebra

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

Step 1: Identify the Key Terms

The phrase 'the quotient of a number and 8' suggests that we are dealing with a division operation. 'Quotient' is a mathematical term that refers to the result of dividing one quantity by another.

Step 2: Define the Unknown Number

We need to use a variable for the unknown number. In this exercise, the variable is given as \( n \).

Step 3: Set Up the Algebraic Expression

Since we want the quotient of the unknown number \( n \) and the number 8, we can express this as \( \frac{n}{8} \). In algebra, division can often be represented using a fraction, where the numerator is the number that is being divided and the denominator is the number by which it is divided.

Key Concepts

Division in Algebra

Division in algebra can sometimes seem tricky, but once you understand the concept, it becomes quite straightforward. Division in algebra is represented by a fraction, where the numerator is the number being divided and the denominator is the number doing the dividing. For example, when we discuss the division of a number by another, we use the word **quotient**. This is a key term that often appears in algebra. In our exercise, we are asked to find the quotient of a number and 8. This simply means we need to divide the unknown number - let's call it 'n' - by 8. Thus, the mathematical expression is set up as \( \frac{n}{8} \). The fraction line is just another way of showing division. Remember, the order of the terms in division matters; the sequence must reflect the problem statement exactly as given.

Variables in Algebra

Variables are fundamental in algebra because they represent unknown values that we can work with in equations. Using variables allows us to solve problems where some information is missing or needs to be found.In our exercise, the variable is represented as \( n \). This is an arbitrary choice, but it's a convention to use letters to represent unknowns. Variables offer a flexible way to express mathematical ideas without knowing every number involved straight away.When you see an expression like \( \frac{n}{8} \), the \( n \) is a placeholder for the number we want to find. As you work through problems, solving for \( n \) will give you specific solutions depending on the rest of the information in your algebraic equation.

Translating Word Problems into Algebraic Expressions

Turning word problems into algebraic expressions is a crucial skill. It allows us to convert everyday language into a mathematical format that can be analyzed and solved. This process starts with identifying key terms and understanding what they represent mathematically.Let's dissect the phrase "the quotient of a number and 8." 'Quotient' is a keyword that hints at division, suggesting we need to divide two numbers. Next, we identify "a number" as our unknown, represented by the variable \( n \). Finally, the number 8 is the divisor.With these pieces, we translate the phrase into the expression \( \frac{n}{8} \). Breaking down problems like this makes them more manageable and shows how mathematics can simplify complex statements into understandable parts.As you practice, always look for these cues—keywords and numbers—and think about how they fit together mathematically.