Problem 62 - Prealgebra

Question

Write the mathematical expressions that are equivalent to each of the following English phrases. Three times the sum of a number and 8

Step-by-Step Solution

Answer

The expression is \( 3(x + 8) \).

Step 1: Identify Key Components

First, identify the key components of the phrase 'Three times the sum of a number and 8'. The key components are the phrases 'three times', 'the sum', 'a number', and '8'.

Step 2: Represent 'a number' as a variable

Choose a variable to represent the unknown number. We typically use the letter \( x \). So, we will let 'a number' be \( x \).

Step 3: Construct the sum inside the expression

The phrase 'the sum of a number and 8' refers to adding the variable \( x \) with 8. This is represented as the expression \( x + 8 \).

Step 4: Apply the multiplication to the sum

The phrase 'three times' means we need to multiply the expression \( x + 8 \) by 3. This results in the expression \( 3(x + 8) \).

Key Concepts

Mathematical Expressions

Mathematical expressions are a way of representing numbers and operations in a compact form. They allow us to express concepts and calculations clearly and concisely. In our exercise, the English phrase 'Three times the sum of a number and 8' needs to be translated into a mathematical expression. This involves identifying operations like addition and multiplication, along with the numbers involved.

• Addition: The phrase refers to 'the sum of a number and 8'. This means we add the number and 8 together, typically represented as \( x + 8 \) in algebraic form.
• Multiplication: The word 'three times' indicates that the sum needs to be multiplied by 3, leading to the expression \( 3(x + 8) \).

By learning to convert words into mathematical symbols, we develop the ability to solve problems in a consistent and logical manner.

Variable Representation

In algebra, variable representation is a critical technique allowing us to generalize and solve problems. Variables serve as placeholders for unknown values or quantities we need to find or manipulate. In our scenario, 'a number' is an unknown quantity. To represent it, we use a variable.
For example, we often use letters like \( x, y, \) or \( z \). In this exercise, we used the letter \( x \) to represent the unknown number. This simplifies complex phrases and makes calculations easy to handle. Using variables enables us to:

• Express relationships between numbers
• Perform arithmetic manipulations
• Formulate general rules or formulas

Variable representation is a fundamental tool in developing equations and solving various mathematical problems.

Step-by-Step Problem Solving

Step-by-step problem solving is an essential skill in mathematics that involves breaking down complex problems into manageable parts. It enhances clarity and ensures that each component of the problem is addressed thoroughly. In our exercise, this approach is illustrated as follows:
1. Identify Key Components: Recognize the operations and numbers in the English phrase.2. Choose a Variable: Assign a letter to represent unknown quantities, simplifying complex expressions.3. Construct the Sum: Form the inner expression based on the wording, such as \( x + 8 \).4. Apply Multiplication: Extend the expression by multiplying the sum by the required number, yielding \( 3(x + 8) \).
This methodical approach ensures that each step logically follows the previous one. It builds a foundation for tackling more challenging problems in the future, emphasizing the importance of understanding each element before moving forward.