Problem 65 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. A number increased by 12

Step-by-Step Solution

Answer

The algebraic expression is \(n + 12\).

Step 1: Interpret the Phrase

The phrase "a number increased by 12" suggests you start with some unknown number, represented by a variable.

Step 2: Identify the Variable

Choose a variable to represent the unknown number. In this case, use the letter \(n\) to denote this number.

Step 3: Translate to Algebraic Expression

The phrase "increased by 12" indicates an addition operation. Therefore, the algebraic expression becomes \(n + 12\).

Key Concepts

Translating Phrases to Expressions

In algebra, we often come across words that need to be converted into mathematical expressions. This process of converting English phrases into algebraic expressions acts like a bridge between verbal and symbolic communication. For example, the phrase, "a number increased by 12," needs to be converted step by step to understand it algebraically.
To begin, identify the key parts of the phrase. The phrase is composed of two main elements: "a number" and "increased by 12." "A number" implies that we are dealing with an unknown value. The term "increased by 12" suggests an increase, which will involve an addition operation. By translating these words into symbols, we present this increase algebraically. It simplifies verbal information into a clear mathematical form that can be easily understood and manipulated.

Unknown Variables

Variables in algebra serve as symbols for unknown or arbitrary numbers. They are placeholders that can represent any number, allowing us to write general expressions and equations. In our example, the unknown number is represented by the variable \(n\).
Choosing a variable is like labeling a blank space with a name. Here, "a number" becomes the variable \(n\). Using variables helps us move from descriptive language to more precise mathematical expressions. It introduces flexibility, allowing the expression to apply to any number, making it a powerful tool in problem-solving. Variables link the unknown to algebra, providing a way to handle calculations without knowing exact values upfront.

Addition Operation

Addition is one of the fundamental operations in arithmetic and algebra. It involves combining values to find their total. In the phrase "a number increased by 12," the core operation is addition.
The word "increased" naturally leads to this operation. When translating to an algebraic expression, you pair the unknown variable (our \(n\)) with the amount it increases by (which is 12) through the addition operation. This results in the expression \(n + 12\).

• The number to be increased is \(n\).
• The amount of increase is 12.
• The resulting expression is \(n + 12\).

This step ensures that the verbal increase is clearly shown in mathematical terms, making calculations straightforward and easy to follow.