Problem 26 - 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 the Variables

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

Step 2: Set Up the Equation

According to the problem, together their total annual salaries sum up to $69,875. Therefore, we set up the equation: \( x + 1.15x = 69,875 \).

Step 3: Combine Like Terms

Combine the terms in the equation: \( 2.15x = 69,875 \).

Step 4: Solve for x

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 Husband's Salary

Perform the division to calculate \( x \): \( x \approx 32,500 \).

Key Concepts

Understanding Variables in Algebra

In algebra, variables play a critical role. They are symbols, often letters like \( x \), used to represent unknown values that we're trying to find. This helps us to write equations that express real-world problems mathematically.

By using a variable, we can denote a value that can change or that we might want to solve for. In the exercise mentioned, the husband's salary is represented by the variable \( x \).

Once we define \( x \) as a variable, we can perform algebraic operations to figure out its value. This allows us to convert complex statements into simple algebraic forms, making the problem easier to solve.

What is a Percentage Increase?

A percentage increase involves finding how much a quantity has grown, expressed as a percent of the original amount. It is a common concept in finance, business, and mathematics.

To determine a percentage increase, you can use the formula:

\[ \text{New Value} = (1 + \text{Percentage Increase as a Decimal}) \times \text{Original Value} \]

For example, if the woman's salary is 15% more than her husband's, it translates to saying her salary is 115% of his. This can be expressed as \( 1.15x \), where \( x \) is the husband's salary. This gives us a straightforward way to express and calculate the incremental increase using algebraic expressions.

Calculation of Salaries Using an Equation

Salary calculation using an equation starts with identifying known and unknown quantities. Here, both combined salaries equal \(69,875, and each has a defined relation through the percentage increase.

To set up the equation, you sum both salaries: the husband's salary \( x \) and the wife's salary, which is 15% more as \( 1.15x \). This gives us the equation: \( x + 1.15x = 69,875 \).

Through solving, you combine like terms to form \( 2.15x = 69,875 \). To isolate \( x \), divide both sides by 2.15:

\[ x = \frac{69,875}{2.15} \approx 32,500 \]

This calculation shows that the husband's salary is approximately \)32,500. The breakdown into components and the structured solving process show how algebraic equations can simplify real-world financial calculations.