Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

A workman's basic hourly wage is \(\$ 10\), but he receives one and a half times his hourly rate for any hours worked in excess of 40 per week. If his paycheck for the week is \(\$ 595\), how many hours of overtime did he work?

Step-by-Step Solution

Answer

The workman worked 13 hours of overtime.

Step 1: Identify Normal and Overtime Hours

First, identify that a regular work week is 40 hours. Any hours worked beyond this are considered overtime hours.

Step 2: Determine Regular Pay

Calculate the weekly wage for regular hours by multiplying the normal hourly rate by 40 hours. \[ \text{Regular Pay} = 10 \times 40 = 400 \]

Step 3: Calculate Overtime Pay

Subtract the regular pay from the total paycheck amount to find the overtime pay. \[ \text{Overtime Pay} = 595 - 400 = 195 \]

Step 4: Calculate Hourly Overtime Rate

Determine the overtime hourly wage, which is one and a half times more than the normal hourly rate. \[ \text{Overtime Rate} = 10 \times 1.5 = 15 \]

Step 5: Calculate Overtime Hours Worked

Divide the overtime pay by the hourly overtime rate to find the number of overtime hours worked. \[ \text{Overtime Hours} = \frac{195}{15} = 13 \]

Key Concepts

Hourly Wage Calculation

Understanding how to calculate an hourly wage is essential for working professionals and students learning financial literacy. An hourly wage represents the amount of money an employee earns for each hour of work performed. In this example, the workman's basic hourly wage is given as \( \\(10 \).
To understand this better, consider the following points:

• Hourly wage helps in calculating the total earnings for a regular workweek, which is typically 40 hours.
• By multiplying the hourly wage by the number of hours worked, you can determine the regular weekly earnings.
• For this exercise, \( 10 \times 40 = \\)400 \), which is the regular pay before any overtime is considered.

Knowing how to calculate hourly wages effectively helps in planning finances and anticipating weekly earnings.

Overtime Rate

An overtime rate is an increased rate of pay for hours worked beyond the standard workweek. Most employees are eligible for this higher payment as a form of compensation for extra work hours.
In this scenario, the overtime rate is calculated as one and a half times the basic hourly wage. With an hourly wage of \( \\(10 \), this means the overtime rate is calculated as follows:

• Multiply the hourly wage by 1.5: \( 10 \times 1.5 = \\)15 \).
• This increased wage helps to incentivize working additional hours beyond the regular 40-hour workweek.
• Knowing how to calculate overtime rates ensures employees are aware of their rights and earnings potential when working beyond their usual hours.

Understanding overtime rates is crucial because it directly affects the total compensation one may receive in a paycheck.

Paycheck Analysis

Analyzing a paycheck involves understanding how your total earnings are broken down between regular and overtime pay. In our example, the workman's total paycheck is \( \\(595 \).
To effectively analyze this paycheck:

• Start by calculating the amount earned from regular hours (40 hours): \( \\)400 \).
• Subtract regular pay from total earnings to find overtime pay: \( 595 - 400 = \$195 \).
• Determine the number of overtime hours worked by dividing the overtime pay by the overtime rate: \( \frac{195}{15} = 13 \) hours.

Paycheck analysis helps workers confirm they have been compensated correctly and gives insight into how much extra is earned during overtime. Having a handle on paycheck components not only aids in personal budgeting but also ensures fairness and transparency in wage calculations.