Problem 53 - Intermediate Algebra

Question

Greg has nickels and pennies in his pocket. The number of pennies is seven less than three the number of nickels. Let \(n\) represent the number of nickels. Write an expression for the number of pennies.

Step-by-Step Solution

Answer

The expression for the number of pennies is \( 3n - 7 \).

Step 1: Define the variable

Let the number of nickels be represented by the variable \( n \).

Step 2: Set up the relationship between nickels and pennies

According to the problem, the number of pennies is seven less than three times the number of nickels. This relationship can be expressed mathematically.

Step 3: Formulate the expression

Translate the relationship into an algebraic expression. Three times the number of nickels can be written as \( 3n \), and seven less than this amount is \( 3n - 7 \).

Step 4: Write the final expression

Combine the earlier steps to express the number of pennies in terms of the number of nickels as \( 3n - 7 \).

Key Concepts

Variable Definition

Understanding variable definition is the first step in solving algebraic expressions. In this exercise, we need to define a variable that will represent the number of nickels.

A variable is simply a symbol or letter that stands in place of an unknown value. Here, we choose the letter \( n \) to represent the number of nickels Greg has. This step is crucial because it helps us translate real-world problems into mathematical language, making it easier to work with and solve.

Relationship Setup

In the next step, we need to establish the relationship between the number of nickels and the number of pennies Greg has.

The problem statement tells us that the number of pennies is seven less than three times the number of nickels. Setting up this relationship correctly is essential because it forms the basis of our algebraic expression.

By carefully reading the problem, we understand that for every number of nickels \(n \), the number of pennies will be calculated based on this relationship.

Expression Formulation

Once we understand the relationship, the next step is to formulate it into an algebraic expression.

We know from the relationship that the number of pennies can be described as three times the number of nickels minus seven. In mathematical terms, 'three times the number of nickels' is written as \(3n \). To account for 'seven less', we subtract seven from \(3n \).

Therefore, our expression for the number of pennies becomes \(3n - 7 \).

Final Expression Writing

The final step is to combine all the earlier steps into one clear algebraic expression.

Starting with our defined variable \( n \), and using the relationship that the number of pennies is seven less than three times the number of nickels, we write:

\[ p = 3n - 7 \]

This expression now allows us to calculate the number of pennies for any given number of nickels Greg has, using a straightforward algebraic equation.

Mastering these steps - from defining variables to writing the final expression - helps simplify complex problems and makes finding solutions easier.