Problem 139 - Introductory Algebra for College Students

Question

Write an algebraic expression for the given English phrase. The value, in cents, of \(x\) nickels

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is \(5x\).

Step 1: Identify the components of the statement

The English phrase mentions two things, the number of nickels represented by 'x' and their total value in cents. We will use this information to write an algebraic expression.

Step 2: Understand the relationship

It's important to remember the value of a nickel, which is 5 cents. Therefore, to find the total value of 'x' nickels, we multiply the number of nickels by 5.

Step 3: Write the algebraic expression

We can write the algebraic expression where \(x\) represents the number of nickels, hence the value in cents is 5 times \(x\) or simply, \(5x\).

Key Concepts

Nickels Value Calculation

Understanding the value of nickels is essential in creating algebraic expressions that reflect their value in cents. Each nickel is worth 5 cents. To find the total value of a certain number of nickels, we need to multiply the number of nickels by their value in cents. For example:

• 1 nickel equals 5 cents
• 2 nickels equal 10 cents
• If you have 10 nickels, their total value is 10 times 5, which is 50 cents

This multiplication tells you how much money in cents the nickels are worth altogether. So, if you have 'x' nickels, you simply multiply 'x' by 5 to find their value, in cents. This leads us to the expression for the total value in cents being represented as \(5x\).

Multiplication in Algebra

Multiplication is a fundamental operation in algebra. It lets us represent repeated addition efficiently, and it's often used when we need to calculate totals. In the context of the exercise, multiplication helps us calculate the total value of nickels by considering how many nickels there are and what each is worth. This allows us to form expressions like \(5x\), where 'x' is the number of nickels. Here are key points to remember about multiplication in algebra:

• It is represented by placing numbers and variables next to each other. For example, \(5x\) means "5 times \(x\)".
• It simplifies calculations. Instead of adding 5 repeatedly, we use multiplication.
• It can be applied generally. Whether you have 1 nickel or 100, the multiplication principle remains the same: \(5 \times\) the number of nickels gives you the total value in cents.

Writing Expressions from Phrases

Translating English phrases into algebraic expressions is a vital skill in mathematics. It involves identifying numerical relationships described in words and converting them into mathematical symbols. Consider the phrase "value, in cents, of \(x\) nickels." Here's how we write it:

• Identify the quantities: "\(x\) nickels" indicates a variable number of nickels expressed as \(x\).
• Determine relationships: A nickel's value is 5 cents, and "value in cents" guides us to multiply. This gives us the relationship described.
• Formulate the expression: Combine these insights to form \(5x\), which directly represents the idea of "\(x\) nickels" times their individual value of 5 cents.

Breaking down sentences this way helps decode word problems and create straightforward algebraic expressions that accurately model real-world situations.