Problem 78 - Algebra for College Students

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Twelve times the sum of a number and 7

Step-by-Step Solution

Answer

The algebraic expression is \(12(n + 7)\).

Step 1: Identify the Unknown

The unknown number in the problem is represented by the variable \(n\). This is the value we need to include in our algebraic expression.

Step 2: Interpret the Sum Part

The phrase "the sum of a number and 7" indicates that we need to add the unknown number \(n\) and 7. This can be written as \(n + 7\).

Step 3: Apply the Multiplication

The phrase "twelve times the sum" tells us to multiply the sum \(n + 7\) by 12. We denote this operation as \(12 \times (n + 7)\).

Step 4: Write the Algebraic Expression

Putting it all together, the algebraic expression for the given phrase is \(12(n + 7)\).

Key Concepts

Variable Representation

In algebra, a variable is a symbol used to represent an unknown or arbitrary number. Variables are often denoted by letters such as \(n\), \(x\), or \(y\). These symbols can stand for any number, and their value isn't fixed.
In the given exercise, the variable \(n\) is used to represent an unknown number.
Using a variable allows for flexibility and makes it possible to form equations and expressions that can model real-world problems or abstract concepts.

• Why use Variables? They simplify complex problems, help in forming equations, and make calculations easier.
• Common Variable Names: Letters such as \(n\), \(x\), \(y\), representing unknowns in algebraic expressions and equations.

Translation of Phrases

Translating verbal phrases into algebraic expressions is a crucial skill in algebra. It involves turning words describing mathematical operations into symbols and equations. In our exercise, the phrase "the sum of a number and 7" converts into an algebraic expression.
Here’s how it works:

• Understanding Key Terms: The word "sum" implies an addition operation. So, whenever you see "the sum of" two numbers in English, you know they need to be added together in the expression.
• Conversion Example: For "a number and 7," you use \(n+7\), adding the variable \(n\) and the constant 7.

Moreover, placing phrases like "twelve times the sum" requires recognizing multiplication processes. In this example, "times" signals multiplication, which further transforms the initially translated sum into a broader operation.

Multiplication in Algebra

Multiplication in algebra is a straightforward yet critical operation. The term "times" in mathematical language signifies multiplication between numbers or expressions. In algebra, you might see expressions like \(a \times b\), which can also be written as \(ab\) or using parentheses, such as \((a)(b)\). In our exercise:

• The phrase "twelve times the sum" translates to a multiplication operation where twelve is multiplied by everything inside the parentheses \((n+7)\).
• We represent this using the expression \(12(n+7)\), meaning that the sum \(n+7\) is to be taken twelve times.

This highlights the use of distributive property when needed and illustrates clear multiplication without ambiguity, ensuring the accurate representation of algebraic operations.