Problem 17 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. one-half of the sum of a number and 4

Step-by-Step Solution

Answer

\( \frac{x + 4}{2} \)

Step 1: Identify the Phrase Components

The phrase given is 'one-half of the sum of a number and 4'. We first identify the key components: 'one-half', 'sum', 'a number', and '4'.

Step 2: Introduce Variables

Choose a variable to represent the unknown number. Let's use \( x \) to represent 'a number'.

Step 3: Express the Sum

Formulate the sum of the number and 4. The sum of a number \( x \) and 4 is expressed as \( x + 4 \).

Step 4: Calculate One-Half of the Sum

To find one-half of this sum, we divide the sum \( x + 4 \) by 2. This is represented as \( \frac{x + 4}{2} \).

Step 5: Final Expression

The algebraic expression for 'one-half of the sum of a number and 4' is \( \frac{x + 4}{2} \).

Key Concepts

Variables

Variables are the building blocks of algebraic expressions. They act as placeholders for unknown values that we aim to discover or work with. In algebra, we often represent variables with letters like \( x \), \( y \), or sometimes even exotic symbols. By assigning variables to unknown numbers, we can simplify complex problems into manageable equations.

In our exercise, the unknown number we want to work with is represented by the variable \( x \). This choice is crucial because it allows us to use algebra to explore the relationship described in the phrase. Whenever we see a term like 'a number' in word problems, it is our cue to introduce a variable. This makes it easier to perform calculations and draw meaningful conclusions about what that number could be.

This approach is very powerful, allowing for flexibility in solving a wide range of problems. Whether you're trying to find the value of a single unknown or working with multiple variables, understanding how to use them effectively is key in algebra.

Arithmetic Operations

Arithmetic operations form the core processes of mathematics, involving addition, subtraction, multiplication, and division. When we translate phrases into algebraic expressions, these operations help us form the correct mathematical representation of the phrase.

In the phrase "one-half of the sum of a number and 4," we encounter several arithmetic operations:

• First, we have the 'sum'. This indicates the operation of addition, specifically between the number represented by the variable \( x \) and 4, resulting in the expression \( x + 4 \).
• Next, 'one-half of' tells us to perform division. We need to divide the sum \( x + 4 \) by 2. This is where division comes into play, giving us the expression \( \frac{x + 4}{2} \).

This sequence of operations illustrates how arithmetic transforms a verbal statement into an algebraic expression, allowing us to calculate or further manipulate the expression for problem-solving.

Translating Phrases into Expressions

Translating phrases into algebraic expressions is like learning a new language, the language of mathematics. It involves carefully identifying key words and phrases that correspond to mathematical operations or structures and expressing these in a concise, symbolic form.

To translate the phrase 'one-half of the sum of a number and 4' into an algebraic expression, we:

• Recognize 'a number' can be represented by a variable, such as \( x \).
• Understand that 'the sum of a number and 4' translates to the operation \( x + 4 \).
• Apply the fraction 'one-half' to the entire sum, indicating division: \( \frac{x + 4}{2} \).

This translation process demonstrates understanding not just of math but also of language. Identifying these components accurately ensures we can represent complex ideas with simple algebraic expressions, paving the way for effective problem-solving in mathematics.