Problem 69 - Algebra for College Students

Question

After 1 yr on the job, Grady got a raise from \(\$ 10.50\) per hour to \(\$ 11.76\) per hour. What was the percent increase in his hourly wage?

Step-by-Step Solution

Answer

The percent increase in Grady's hourly wage is 12 percent.

Step 1 - Identify the initial and new wages

The initial wage is \(\$10.50\) per hour, and the new wage is \(\$11.76\) per hour.

Step 2 - Calculate the difference in wages

Subtract the initial wage from the new wage to find the difference: \(\$11.76 - \$10.50 = \$1.26\).

Step 3 - Determine the percent increase

To find the percent increase, divide the difference by the initial wage and multiply by 100: \(\frac{1.26}{10.50} \times 100 = 12\) percent.

Key Concepts

Percent Change

Understanding percent change is essential for many real-world applications.
Specific to Grady's situation, we are looking at the percent increase in his wage.
To determine this, we follow a few basic steps:

• Identify the initial quantity (initial wage) and the new quantity (new wage).
• Calculate the difference between the two quantities.
• Divide the difference by the initial quantity.
• Multiply the result by 100 to get the percentage.

In mathematical terms, the formula looks like this:
\[ \text{Percent Increase} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \ \text{Percent Increase} = \frac{11.76 - 10.50}{10.50} \times 100 = 12\text{\text{ percent}}\]Applying these steps to Grady’s raise, we get a 12 percent increase in his hourly wage. It's important to practice with different scenarios to become comfortable with calculating percent changes.

Wage Calculation

Wage calculation is crucial for budgeting and financial planning.
It involves understanding both your current earnings and any changes, like raises.
For Grady, knowing his old and new wages allows him to determine the increase and plan accordingly. If Grady works 40 hours a week, his new weekly wage can be calculated easily:

• Initial weekly wage: \(10.50 \times 40 = 420\) dollars
• New weekly wage: \(11.76 \times 40 = 470.40\) dollars

Annual wage can also be calculated to understand yearly earnings better.
Thus, Grady’s annual wage before the raise was \(420 \times 52 = 21,840\) dollars, and after the raise, it is \(470.40 \times 52 = 24,460.80\) dollars. Understanding these calculations helps one to make informed decisions about savings and expenses.

Algebra Problem

Many problems, like wage calculations, can be solved using algebra.
Algebra provides a systematic way to handle equations and unknowns. Using Grady's wage increase as an example:

• Let's denote the old wage as \( W_1 = 10.50\)
• The new wage as \( W_2 = 11.76\)

The change in wage, \( \triangle W \), is \(W_2 - W_1\)
For Grady, this is \(11.76 - 10.50 = 1.26\)

To find the percent increase, we set up the following equation:
\[ P = \frac{\triangle W}{W_1} \times 100 \ Percent\underline{\phantom{xxx}} Increase = \frac{1.26}{10.50} \times 100 = 12\text{ percent}\]By substituting into this formula, we simplify the calculation process.
Through algebra, such problems become easy to solve, supporting an organized approach to mathematical reasoning.