Problem 77 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Ten times the difference of a number and 6

Step-by-Step Solution

Answer

The algebraic expression is \(10(n - 6)\).

Step 1: Identify the Components of the Phrase

The phrase "Ten times the difference of a number and 6" consists of two main components: a number (which is represented by the variable \(n\)) and the difference between this number and 6.

Step 2: Translate 'the difference of a number and 6'

The difference between the unknown number \(n\) and 6 is represented algebraically as \(n - 6\). This expression captures the meaning of 'the difference of a number and 6'.

Step 3: Translate 'Ten times' into a Mathematical Operation

The phrase 'ten times' indicates that we need to multiply a given expression by 10. Hence, we multiply the previously found expression \(n - 6\) by 10.

Step 4: Formulate the Full Algebraic Expression

Combine the results from the previous steps to form the expression. The full algebraic expression for 'Ten times the difference of a number and 6' is \(10(n - 6)\).

Key Concepts

Translating Phrases to Expressions

Turning phrases into algebraic expressions is like deciphering a secret code. It's all about understanding the relationships between words and numbers. When you read phrases like "the sum of a number and three" or "twice a number," you're actually being given instructions on how to create an algebraic expression. Let's break this down:

• Identify Key Words: Words like "sum," "difference," "product," and "quotient" are math operations in disguise. They tell you what operation to use in your expression.
• Find the Main Subject: Often, the phrase will revolve around an unknown number, which we can represent with a variable, such as \(n\).
• Build the Expression: Once you know the math operation and the main subject, you can assemble these parts into an algebraic expression that mirrors the phrase.

By translating each piece of the phrase into mathematical language, you're able to construct an expression that accurately represents the given situation.

Unknown Variables

In algebra, a variable acts as a placeholder for unknown values. It's usually represented by a letter, like \(n\), \(x\), or \(y\). These symbols stand in for numbers or quantities we don't yet know.Variables are powerful because:

• Versatility: They can represent any number, making expressions flexible and widely applicable.
• Problem Solving: Variables allow you to write equations that express relationships between different quantities, helping to solve complex problems.
• Simplification: Instead of dealing with cumbersome real-world numbers, variables let you work out solutions more easily and reveal patterns.

In the exercise example, \(n\) is used to symbolize an unknown number that interacts with known numbers through mathematical operations. This approach lets us think about general solutions rather than being limited to specific numeric examples.

Mathematical Operations

Mathematical operations are the backbone of constructing and understanding algebraic expressions. There are four basic operations that help manipulate numbers and variables: addition, subtraction, multiplication, and division. Here's how they play out:

• Addition (+): Combining numbers or variables to get a sum. For example, \(n + 3\) means adding 3 to \(n\).
• Subtraction (−): Finding the difference between numbers or variables. As in \(n - 6\), where 6 is subtracted from \(n\).
• Multiplication (×): Scaling numbers or variables. So, "ten times a difference" means multiplying that difference by 10, like in \(10(n - 6)\).
• Division (÷): Splitting numbers or variables into equal parts, less common in simple expressions but crucial in more complex ones.

In the phrase "Ten times the difference of a number and 6," we are instructed to subtract (find a difference) and then multiply (find a product). Each operation should be applied carefully to maintain the structure of the original phrase.