Problem 81 - Intermediate Algebra

Question

Answer the question with an algebraic expression. Pam is \(t\) years old, and her mother is 3 less than twice as old as Pam. What is the age of Pam's mother?

Step-by-Step Solution

Answer

Pam's mother's age is \(2t - 3\).

Step 1: Identify Pam's Age

Understand that Pam's age is given by the variable \(t\). This is a starting point for forming an expression for Pam's mother's age.

Step 2: Understand the Mother's Age in Relation to Pam's Age

Note that Pam's mother is described as being 3 years less than twice Pam's age. Mathematically, this can be translated into: 2 times Pam's age (\(2t\)) minus 3.

Step 3: Formulate the Expression for Mother's Age

Combine the information from Step 2 to create an algebraic expression for Pam's mother's age: \(2t - 3\).

Key Concepts

Understanding Age Problems in Algebra

Age problems in algebra are common scenarios where we use mathematical expressions to represent people's ages at different times. These problems help us understand how to translate real-world situations into algebraic language, allowing for precise solutions.

Age problems usually involve comparing the ages of two or more individuals.To solve these problems:

• Identify the ages using variables.
• Describe the relationships between the ages using expressions or equations.
• Apply the information given to create mathematical forms.

In our exercise, Pam's age is expressed with the variable \(t\), which is a simple yet powerful representation. Understanding these age problems helps build a strong foundation for more complex algebraic expressions.

The Role of Variables in Algebra

Variables are symbols, often letters like \(t\), used to represent unknown quantities.They are fundamental in algebra as they allow us to write general formulas and expressions that can solve numerous problems.

In the context of age problems, variables help in representing ages that can change and are unknown fixed numbers at a point. For our given problem:

• \(t\) stands for Pam's current age.
• It allows us to express other related ages, such as her mother's age, using mathematical expressions.

Variables make it possible to manipulate expressions and to solve equations that represent real-life situations. Their use helps in maintaining a clear and structured approach to problem-solving.

Forming Expressions for Age Calculations

Forming expressions is at the core of solving algebraic problems like those involving age.This involves converting word statements into mathematical language, which is more precise and unambiguous.

In our example, we use Pam's age \(t\) to find her mother's age based on a description:

• Pam's mother is 3 less than twice Pam's age.
• This translates to an expression \(2t - 3\), where \(2t\) represents twice Pam’s age and we subtract 3 to adjust the given condition.

Creating such expressions allows us to easily solve for unknown values once more specific data is given.This skill is essential for anyone studying algebra and applying it to practical problems.