Problem 30 - Elementary Algebra Within Reach

Question

Salary Raise A union negotiates for a cost-of-living raise of \(4.5 \%\). What is the raise for a union member whose salary is \(\$ 37,380\) ? What is this person's new salary?

Step-by-Step Solution

Answer

The raise comes out to be $1682.10. Therefore, the new salary would be $39,062.10.

Step 1: Calculate the raise

To find the raise, a simple percentage calculation can be performed. Multiply the original salary by the increase percentage divided by 100. Here, \(4.5 \%\) of \(\$37,380\) is equivalent to \(\$37,380 \times \frac{4.5}{100}\)

Step 2: Calculate new salary

Once the raise is determined, the new salary can be calculated by summing the original salary and the raise. Thus the new salary is the sum of \(\$37,380\) and the calculated raise.

Key Concepts

Percentage Calculation

Understanding percentage calculation is essential when determining salary raises, discounts, or interest rates. A percentage represents a fraction of 100, and to calculate a percentage of a given amount, you simply multiply the amount by the percentage expressed as a decimal. For example, if you're to calculate a 4.5% raise on a salary, the percentage in decimal form is 0.045 (since 4.5% equals 4.5/100 or 0.045).

To find the raise, the salary (\f\(37,380) is multiplied by this decimal: \f\)37,380 \times 0.045. This calculation will give you the amount of money that the 4.5% raise represents. It's a straightforward process that applies to a variety of financial scenarios, providing a clear picture of increments or decrements expressed in percentages.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (representing numbers we don't yet know), and operation symbols. In the context of calculating a salary raise, the algebraic expression would be the formula used to find the raise amount. For instance, if 's' represents the initial salary and 'r' represents the raise percentage converted into a decimal, the raise can be calculated using the expression 's * r'.

In our example, 's' is \f\(37,380, and 'r' is 0.045 (as 4.5% raise). The algebraic expression to calculate the raise would be \f\)37,380 * 0.045, which gives the exact amount of the raise. Using algebraic expressions helps simplify and solve problems step by step in a logical and consistent manner, making it a powerful tool in various applications beyond salary calculations.

Cost-of-Living Adjustment

A cost-of-living adjustment, or COLA, is an increase in income to maintain the purchasing power of an individual in the face of inflation. It's usually a percentage based on the rise in consumer price indices and is often applied to salaries, pensions, and benefits. For a union member, a COLA ensures that their income corresponds with increased living costs without diminishing their standard of living.

In our exercise scenario, the union negotiated a 4.5% COLA for its members. This raise helps the workers sustain their purchasing power in a growing economy where prices for goods and services might be rising. The practical impact of a COLA can be understood not just as a salary bonus, but as a necessary adjustment to maintain financial stability in an ever-changing economic environment.