Problem 139 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(139-142\), 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 'The value, in cents, of \(x\) nickels' is \(5x\).

Step 1: Interpret the Given Phrase

The English phrase 'The value, in cents, of \(x\) nickels' translates to 'how much the total value would be if each nickel (which is \(5\) cents) is multiplied by \(x\)'.

Step 2: Convert to Algebraic Expression

In terms of mathematics, we multiply to find out the total. Here, \(x\) represents the number of nickels, and the value of each nickel is \(5\) cents. The total value, in cents, of \(x\) nickels will be \(5x\).

Key Concepts

Translating English Phrases to Algebra

Converting English sentences into algebraic expressions is a fundamental skill in algebra. It helps in forming equations that can be solved to find desired values. In this context, let's understand the process through the example of determining the value of "x nickels."

The given phrase is "The value, in cents, of \(x\) nickels." To start translating, we need to identify what each word represents in terms of numbers and operations. The word "value" suggests we are looking to determine a numerical worth. "Nickels" are coins, each worth \(5\) cents. The phrase "of \(x\) nickels" indicates that this value is repeated or multiplied \(x\) times.

This English phrase converts into the algebraic expression \(5x\). Here, \(x\) symbolizes the number of nickels, and by multiplying \(x\) by \(5\), we get the total value in cents.

Monomials

A monomial is an algebraic expression that consists of only one term. It can be a constant, a variable, or a product of constants and variables. In the expression \(5x\), each component contributes to it being classified as a monomial:

• Constant: The number \(5\) is a constant or a fixed value.
• Variable: \(x\) represents a changeable element, like the number of nickels.
• Product: The combination of the constant and variable \(5x\) represents the product of the two, forming a single term.

Monomials are the simplest type of algebraic expressions, and operations such as addition, subtraction, multiplication, and division can be performed on them. Recognizing a monomial helps in managing these operations and understanding more complex expressions.

Value of Coins

Understanding the value of different coins is pivotal in solving various mathematical problems. In this exercise, the focus is on nickels, each worth \(5\) cents. To calculate the value of several nickels, we multiply the number of nickels by their value.

Here are key points about calculating the value of coins:

• Identify Coin Type: Determine what kind of coin you are dealing with (nickels, dimes, quarters, etc.). Each type has a specific value.
• Value per Coin: Know the value of each coin type. For nickels, it's \(5\) cents.
• Total Coins: Figure out how many coins are there. This is denoted by a variable like \(x\).
• Calculate Total Value: Multiply the number of coins by the value of each coin. For example, with nickels, the calculation is \(5x\).

These steps provide a systematic approach to finding out the total worth of a collection of coins, making it easier to manage and solve everyday math problems involving money.