Problem 41 - Algebra for College Students

Question

Solve each equation. Mitsuko's salary for next year is \(\$ 34,775\). This represents a \(7 \%\) increase over this year's salary. Find Mitsuko's present salary.

Step-by-Step Solution

Answer

Mitsuko's present salary is $32,500.

Step 1: Understanding the Problem

We know that Mitsuko's next salary is $34,775, which includes a 7% raise from her current salary. Let the current salary be denoted by \(x\). Our task is to find the original salary before the increase.

Step 2: Setting Up the Equation

The relationship between the next year's salary and this year's salary can be written as \(x + 0.07x = 34,775\). This simplifies to \(1.07x = 34,775\).

Step 3: Solving for Present Salary

To find \(x\), divide both sides of the equation by 1.07: \[ x = \frac{34,775}{1.07} \].

Step 4: Calculating the Result

Perform the division to compute \(x\). This results in \( x = 32,500 \).

Step 5: Conclusion

Mitsuko's present salary is \(\$32,500\).

Key Concepts

Understanding Percentage Increase

When we talk about percentage increase, we're referring to the amount something grows compared to its original size. In this problem, Mitsuko received a 7% salary increase. This means that her new salary is 107% of her previous one. To clarify:

• The original value is considered 100%.
• A 7% increase adds 7% more.

Thus, her new salary is equivalent to 107% of what she previously earned. Mathematically, the relationship can be expressed as:\[ \text{New Salary} = 1.07 \times \text{Current Salary} \]Understanding this is crucial as it helps set up the equation needed to find Mitsuko's current salary.

Forming the Linear Equation

Linear equations are equations of the first order. They deal with relationships that can be graphed as a straight line. In the context of this problem, we use a linear equation to express the relationship between Mitsuko's current and next salary. We denote her current salary by \(x\). The equation is:

• Current Salary: \(x\)
• Next Year's Salary: \(x + 0.07x = 1.07x\)

This equation states that Mitsuko's upcoming salary is her current salary plus 7% of it. Recognizing this as a linear equation allows us to solve for \(x\) systematically.

Solving the Equation

To solve for Mitsuko’s current salary \(x\), we need to tackle the equation \(1.07x = 34,775\). This involves isolating \(x\), which is achieved by dividing both sides of the equation by 1.07:\[ x = \frac{34,775}{1.07} \]This operation effectively "undoes" the multiplication by 1.07, leaving us with Mitsuko's current salary. Performing the division gives:\[ x \approx 32,500 \] This tells us Mitsuko’s current salary before the increase was approximately $32,500. Solving equations in this way involves systematic operations to isolate the variable, which allows us to determine the unknown value.