Problem 18 - Algebra and Trigonometry

Question

Salaries A woman earns 15\(\%\) more than her husband. Together they make \(\$ 69,875\) per year. What is the husband's annual salary?

Step-by-Step Solution

Answer

The husband's annual salary is approximately $32,500.

Step 1: Define Variables

Let the husband's annual salary be denoted by \( x \). Since the woman's salary is 15\( \% \) more than her husband, her salary can be expressed as \( 1.15x \).

Step 2: Set Up the Equation

Together, their total annual salary is \( 69,875 \). Therefore, we can write the equation: \[ x + 1.15x = 69,875 \]

Step 3: Combine Like Terms

Combine the terms for \( x \) in the equation: \[ 2.15x = 69,875 \]

Step 4: Solve for the Husband's Salary

To find the husband's salary \( x \), divide both sides of the equation by 2.15: \[ x = \frac{69,875}{2.15} \]

Step 5: Calculate the Value

Compute the division to find \( x \): \( x \approx 32,500 \). Therefore, the husband's annual salary is approximately \$32,500.

Key Concepts

Equations

Equations are one of the fundamental building blocks of algebra. They provide a way to find unknown values when other information is known. In this problem, the equation we used was: \[ x + 1.15x = 69,875 \] This equation represents a real-world scenario involving salaries. The husband's annual salary is denoted by \( x \), and his wife's salary is expressed as \( 1.15x \), meaning it includes an additional 15\(\%\) on top of his salary. The goal is to solve for the value of \( x \), which represents the husband's salary. By setting up an equation, you can use algebra to resolve the unknown based on a total known value.

Variables

Variables in algebra are symbols, usually letters, that represent numbers we do not yet know. In our exercise, the variable \( x \) stands for the husband's salary. Variables are useful because they allow equations to be formulated and calculations to be performed to find these unknowns.

• Use variables to represent quantities that can change or vary.
• Substitute the values back into the equation to verify solutions.

By thinking of \( x \) as an empty placeholder, we focus on applying arithmetic to uncover its value, thus transforming the problem into a manageable task.

Percentages

Percentages are a way of expressing a number as a fraction of 100. They're crucial in scenarios like our exercise where comparisons between values are needed. The woman's salary is said to be 15\(\%\) more than her husband's. In mathematical terms, this means: \[ \text{Woman's Salary} = \text{Husband's Salary} + 0.15 \times \text{Husband's Salary} \] Which simplifies to: \[ 1.15x \] Percentages help to easily understand how one quantity relates to another and are fundamental in financial calculations, such as salary comparison and growth rates.

Problem-Solving

Problem-solving in mathematics is the process of finding solutions to complex or simple issues by applying mathematical knowledge. This exercise involves a few steps to correctly identify and solve the problem:

• Understand what is being asked. We need to find the husband's salary.
• Translate the situation into an equation using variables and percentages.
• Solve the equation systematically by combining like terms and using arithmetic operations such as division.
• Check if the solution is reasonable by considering the context.

By approaching problems step-by-step, you demystify them and make them more approachable, enhancing both your understanding and your confidence in tackling mathematical challenges.