Problem 95 - Intermediate Algebra

Question

Answer the question with an algebraic expression. Mila's monthly salary is \(d\) dollars. What is her annual salary?

Step-by-Step Solution

Answer

Mila's annual salary is represented by \(12d\).

Step 1: Understand the Problem

The problem provides Mila's monthly salary, which is given as the variable \(d\) dollars. We need to find her total earnings over one year, which consists of 12 months.

Step 2: Identify the Formula

To find the annual salary, we need to multiply the monthly salary by the number of months in a year. The number of months in a year is 12.

Step 3: Set Up the Expression

Use the information to set up an expression: Annual Salary = Monthly Salary \(\times\) 12. In mathematical terms, this can be written as:\[\text{Annual Salary} = d \times 12\]

Step 4: Simplify the Expression

Simplify the expression for clarity. Since the multiplication can be done directly, the simplified expression for Mila's annual salary is:\[12d\]

Key Concepts

Understanding Variables in Algebraic Expressions

In algebra, a variable represents an unknown or changeable quantity, typically denoted by a letter such as \(x\), \(y\), or \(d\). In our exercise, Mila's monthly salary is expressed by the variable \(d\). This means that instead of using a fixed number for her salary, we use \(d\) to denote that the actual amount can vary or be replaced with different numerical values later.

Variables have several key roles in mathematics:

• They allow for generalization and abstraction, making it possible to create formulas applicable to many situations.
• They serve as placeholders in equations and expressions, assisting in problem-solving by maintaining the setup before plugging in actual numbers.
• They help in spotting patterns and making connections between various mathematical concepts.

Using variables like \(d\) makes calculations and resulting formulas flexible and applicable to real-world situations, such as different salary amounts for different individuals.

The Role of Multiplication in Algebra

Multiplication is a fundamental arithmetic operation, crucial for solving algebraic expressions. It involves combining groups of equal sizes and is symbolized by \( \times \) or sometimes just by juxtaposing numbers and variables.

In our problem, to determine Mila's annual salary from her monthly salary, multiplication is used to calculate how much she earns over 12 months. Here's how it works:

• The number 12 represents the months in a year, a constant that we multiply with \(d\), the variable for her monthly salary.
• Performing the multiplication directly results in the expression \(12d\), combining the number of months and the monthly salary into one compact representation for the annual earnings.

Multiplication helps in scaling numbers, which is especially useful in contexts like finances where quantities need to be repeatedly added over time, as seen in this exercise.

Problem Solving Using Algebraic Expressions

Problem-solving with algebraic expressions often involves setting up and simplifying expressions based on given information. At the heart of it is understanding what the problem asks and then translating words into algebraic symbols and operations.

Consider our exercise where the objective is to find Mila's annual salary. Here's the problem-solving process:

• Identify what you know: Mila's monthly salary is \(d\).
• Understand what you need: Her total earnings over a year.
• Use mathematics to form a strategy: Multiply her monthly salary by 12 months to find the annual salary.

This structured approach of defining the problem, setting up an algebraic expression, and performing calculations can be applied to numerous problems beyond this example. It enables breaking down complex situations into manageable and solvable parts.