Problem 29 - Elementary Algebra Within Reach

Question

Salary Raise You accept a job with a salary of \(\$ 35,600\). After 6 months, you receive a \(5 \%\) raise. What is your new salary?

Step-by-Step Solution

Answer

The new salary after a 5% pay raise will be \$ 37,380.

Step 1: Express the Percentage Increase as a Decimal

A 5% increase is equivalent to \(0.05\) in decimal form. We need to convert the percentage to a decimal, because we will be performing multiplication operations, which are easier to do with decimals than with percentages. So, 5% = \(0.05\).

Step 2: Add One to the Decimal Equivalent of the Percentage Increase

The next step is to add \(1\) to the decimal equivalent of the percentage increase. This transforms the increase into a multiplication factor that we can apply directly to the initial salary. So, \(1 + 0.05 = 1.05\).

Step 3: Calculate the New Salary

The new salary can be found by multiplying the initial salary by \(1.05\). This is equivalent to calculating a 5% increase of the initial salary. So \(35600 * 1.05 = \$ 37,380\).

Key Concepts

Decimal Conversion

Understanding decimal conversion is crucial for performing precise calculations. When dealing with percentage increases, converting percentages to decimals makes the process smoother. Here's how it works.
Suppose you have a percentage, like 5%. To convert it to a decimal, you divide by 100. This is because percent means "per one hundred."
For example:

• 5% becomes \( \frac{5}{100} = 0.05 \).

This conversion is a vital step when you need to use percentages in calculations, especially in operations like multiplication.

Salary Calculations

Calculating salary adjustments is a common real-world application of math. It's essential to understand how a raise affects your salary.
After converting a percentage increase to a decimal, you use it to find your new salary by adding it to 1.
Here’s the reasoning:

• Your raise is not just 5%, it's 105% of the original amount.
• This is because you keep your original salary and add an extra 5%.
• Thus, multiplying by 1.05 makes the calculation easy: \( 35600 \times 1.05 = 37380 \).

This formula helps you effortlessly determine the new salary after any percentage increase.

Algebraic Problem-Solving

Algebraic problem-solving provides tools to tackle various real-life situations using math. In salary calculations, it helps structure our approach to finding solutions.When faced with a problem like calculating a salary raise, break it down:

• First, understand what you know (initial salary, raise percentage).
• Convert necessary values (percentage to decimal).
• Then, perform the calculation using the formula: new salary = initial salary \( \times (1 + \text{decimal raise}) \).

These steps help systematically solve the problem while ensuring accuracy.