Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Weekly Salary In Exercises 57 and 58 , use the following information to write a mathematical model and solve. Due to economic factors, your employer has reduced your weekly wage by \(15 \%\). Before the reduction, your weekly salary was \(\$ 425 .\) What is your reduced salary?

Step-by-Step Solution

Answer

The reduced salary is $361.25. So after a 15% reduction from the original $425 weekly salary, the new salary is $361.25 per week.

Step 1: Understand the Problem

You're given that the salary reduction is 15% and the original salary is $425. The task is to find the reduced salary.

Step 2: Calculate the Reduction

When there's a reduction of 15%, it means that the new salary will be the remaining 85% of the original amount. To find this value, we'll multiply the original salary with 85% (or 0.85 in decimal form).

Step 3: Apply the Reduction to the Original Salary

Multiply $425 by 0.85. This means we are finding 85% of the original salary, which gives us the reduced salary.

Key Concepts

Understanding Percent Decrease

In college algebra, understanding percent decrease is crucial when analyzing changes in quantities such as salaries, prices, or other numerical values. A percent decrease represents a reduction in value by a specific percentage.

• To calculate a percent decrease, you first identify the percentage by which the original amount is reduced.
• Next, convert the percentage to its decimal form by dividing by 100.
• Multiply the original value by this decimal to find the amount reduced.
• Finally, subtract this reduced amount from the original value to find the new decreased amount.

In our example, a salary is decreased by 15%. This translates to retaining only 85% of the original amount. Comprehending these steps helps simplify problem-solving processes related to percentage changes.
Practicing these calculations with various examples can enhance your proficiency in solving real-world problems involving percent decreases.

Mathematical Modeling for Problem Solving

Mathematical modeling is a powerful tool that assists in understanding and solving real-world problems. It involves creating a mathematical representation of a situation, simplifying complex problems into more manageable forms.
When faced with a problem like determining a reduced salary, a model can help you organize the information clearly:

• Identify the knowns: The original salary and the percentage decrease.
• Determine the unknown: The new, reduced salary.
• Establish relationships: The reduced salary is the original salary minus the amount it decreases.

In the provided example, we model the problem by calculating the remaining portion after the percentage decrease. This ensures a systematic approach that can be applied to varied scenarios, reinforcing understanding through consistent structure.

Problem Solving Steps in Mathematics

Effective problem solving in mathematics requires a clear sequence of steps to ensure that each part of a problem is thoroughly addressed. For instance, in calculating a reduced salary due to a percent decrease, the steps are:
- **Step 1: Understand the Problem** - Carefully read and comprehend what is being asked. - **Step 2: Calculate the Relevant Values** - Identify the percentage reduction and convert it to a decimal. - Use this to find how much is left of the original quantity (e.g., 85% in the salary example). - **Step 3: Apply the Calculation** - Multiply the original value by the percentage left (in decimal form) to determine the new value.
Following a problem-solving strategy not only aids in reaching the correct answer but also enhances the ability to tackle similar problems in future scenarios. This planned approach strengthens logic and analytical skills, fostering confidence in mathematical endeavors.