Problem 71 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Four less than one-half of a number

Step-by-Step Solution

Answer

\( \frac{1}{2}n - 4 \)

Step 1: Identify the Unknown

The problem states 'a number,' which is unknown. We will represent this number with the variable \( n \).

Step 2: Translate 'One-Half of a Number'

The phrase 'one-half of a number' can be interpreted mathematically as \( \frac{1}{2}n \). This means multiply the number \( n \) by \( \frac{1}{2} \).

Step 3: Interpret 'Four Less Than'

The phrase 'four less than' suggests subtraction. To translate it into an expression, we take a quantity and subtract 4 from it.

Step 4: Combine the Mathematical Expressions

To find 'four less than one-half of a number,' we take \( \frac{1}{2}n \) and subtract 4 from it. Thus, the algebraic expression becomes \( \frac{1}{2}n - 4 \).

Key Concepts

Translating Phrases into Algebraic Expressions

Understanding how to translate verbal phrases into algebraic expressions is key to mastering basic algebra. The goal is to transform a sentence or phrase involving numbers into a mathematical form that is easy to work with. Let's consider the phrase "four less than one-half of a number," which is part of our exercise.We start by identifying key components of the phrase:

• "One-half of a number"
• "Four less than"

These components guide us on how to translate the phrase into an algebraic expression. "One-half of a number" suggests multiplying an unknown number by 0.5, which can be represented as \( \frac{1}{2}n \). Next, "four less than" indicates a subtraction operation. Thus, subtract 4 from the previously identified expression \( \frac{1}{2}n \). This gives the algebraic expression \( \frac{1}{2}n - 4 \).

This translation process involves stringing together smaller translated pieces from the phrase into a complete expression. Practice and familiarity with common terms like "less than," "sum," and "product" can help you get better at this.

Variables in Algebra

Variables are symbols, like letters, used to represent unknown or arbitrary numbers. In algebra, we frequently encounter the use of variables to help solve problems. They are placeholders for values we either don't know yet or need to find.In our exercise, the variable is \( n \), a standard choice for denoting an unknown number. Using a variable allows us to formulate expressions and equations that can then be solved. For instance, if we were solving an equation based on our expression \( \frac{1}{2}n - 4 \), the variable \( n \) would stand for the number that makes the equation true. This aids in generalizing mathematical statements and performing various mathematical operations.

Variables are not just limited to letters like \( n \). You could use any letter or symbol, but the common ones like \( x \), \( y \), and \( z \) are typically used. This concept of variables is foundational, as it sets the stage for algebraic manipulation, allowing expressions to be simplified and solved effectively.

Basic Algebra Steps

Breaking down complex problems into basic algebra steps makes solving them manageable. Accurately following these steps is crucial in finding the solution to algebraic problems.Here are simple steps usually involved:

• Identify the Unknown: Determine what the problem is asking you to find. Choose a variable to represent this unknown.
• Translate Components: Break down the given phrase into parts that can be written as mathematical terms. Each part of the phrase corresponds to an operation or expression. For example, "one-half of a number" becomes \( \frac{1}{2}n \).
• Formulate the Expression: Combine your translated parts into a coherent mathematical expression or equation. Here, "four less than" suggests subtraction, leading to the formation \( \frac{1}{2}n - 4 \).

Following these steps allows you to convert verbal phrases into algebraic expressions systematically. As you become more practiced, recognizing patterns and translating will become second nature.