Problem 69 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. The product of a number and 50

Step-by-Step Solution

Answer

The expression is \( 50n \).

Step 1: Identify the Operation

The phrase "the product of" indicates multiplication. In this problem, you need to calculate the product of a number and 50.

Step 2: Choose the Variable

The unknown number is represented by the variable \( n \). Thus, you will use \( n \) in the algebraic expression.

Step 3: Construct the Algebraic Expression

To form the expression, multiply the variable \( n \) by 50. The expression is written as \( 50n \).

Key Concepts

Translation of Phrases into Algebra

In algebra, words and phrases in mathematics can be translated into algebraic expressions to solve problems more effectively. When encountering a phrase like "the product of a number and 50," it's crucial to identify keywords that indicate mathematical operations. Here, "the product of" is a key phrase that signifies multiplication.

• "The product of" translates to multiplication.
• "A number" is often the unknown variable we want to find or work with.
• The given numerical value, in this case, is 50.

By understanding these terms, we can translate word problems from English into the language of algebra, allowing us to write equations or formulas that can be solved or simplified. This skill is fundamental in moving from simple arithmetic to more complex algebraic equations.

Use of Variables

Variables are one of the building blocks of algebra. They are symbols used to represent unknown or changeable values within mathematical expressions or equations. In the given problem, the letter \( n \) is used as a variable to denote an unknown number. Choosing the correct variable:

• It's common practice to use lowercase letters, such as \( x \), \( y \), or \( n \), to represent variables.
• The variable should be clearly defined and consistent throughout the problem.

Using a variable allows flexibility in algebra. It not only stands for a single number but can represent different values depending on the context of the problem. Introduces a way to easily construct and manipulate algebraic expressions without specifying a number until needed.

Multiplication in Algebra

Multiplication in algebra retains much of the simplicity from basic arithmetic but applies it within algebraic expressions involving variables. In the exercise, the phrase "the product of a number and 50" requires translation to an expression using multiplication.Here's how multiplication works in algebra:

• Instead of writing out the multiplication operation in full, use juxtaposition (placing numbers and variables next to each other). For example, \( 50n \) means \( 50 \times n \).
• Multiplication is commutative, meaning the order doesn't matter: \( 50n = n \times 50 \).

By following these conventions, algebraic multiplication helps in constructing neat, efficient expressions. This allows for easier simplification and evaluation in more advanced algebraic tasks.