Problem 60 - Prealgebra

Question

Suppose a woman makes \(\$ 286.08\) in one week. If she is paid \(\$ 5.96\) an hour for the first 36 hours she works and then \(\$ 8.94\) an hour in overtime pay for each additional hour she works in the same week, how many hours did she work overtime that week?

Step-by-Step Solution

Answer

The woman worked 8 hours of overtime that week.

Step 1: Calculate Regular Earnings for the First 36 Hours

First, calculate how much the woman earned for the first 36 hours at her regular rate of pay. Her regular hourly rate is \( \\(5.96 \). Thus, her earnings for 36 hours are computed as:\[ 36 \times 5.96 = 214.56 \]Therefore, she earned \\)214.56 for the first 36 hours.

Step 2: Determine Earnings from Overtime

Subtract the regular earnings from her total weekly earnings to find out how much she earned from overtime.\[ 286.08 - 214.56 = 71.52 \]So, she earned \$71.52 from overtime work.

Step 3: Calculate Number of Overtime Hours Worked

Given her overtime pay rate is \( \$8.94 \) per hour, we need to calculate the number of overtime hours. Using the formula for total overtime pay:\[ \text{Overtime Hours} = \frac{\text{Overtime Pay}}{\text{Overtime Rate}} = \frac{71.52}{8.94} \approx 8 \]So, she worked approximately 8 hours of overtime.

Key Concepts

Hourly Wage Calculation

When calculating pay, the hourly wage is the foundation. It determines how much you earn for every hour of work. In this scenario, the woman's regular hourly wage is \(\\(5.96\) for her first 36 hours each week. This is the amount agreed upon as her base rate for what is considered her standard workweek.

To find out what she earns in a standard 36-hour week, you multiply the number of hours (36) by the hourly rate. Here's the calculation:

• Regular Weekly Hours = 36
• Regular Hourly Wage = \(\\)5.96\)
• Earnings from Standard Hours = \(36 \times 5.96 = 214.56\)

Understanding your hourly wage helps you manage your finances better. It's essential to know how much you make before overtime or additional bonuses. This base is crucial for planning and budgeting.

Earnings Calculation

Earnings calculation involves summing all sources of income in a given period. For the woman in this problem, we first calculated her earnings from her regular hours. Next, we need to include her overtime earnings to determine her total weekly pay.

The total earnings she's entitled to for the week are \(\\(286.08\). After calculating her regular hourly earnings, we subtract this from her total earnings to check what amount comes from overtime. This tells us:

• Total Earnings = \(\\)286.08\)
• Regular Earnings = \(\\(214.56\)
• Earnings from Overtime = \(\\)286.08 - \\(214.56 = \\)71.52\)

This breakdown is critical because it clearly shows how much was earned at the standard rate versus how much resulted from additional work. Understanding each component of your earnings helps in accurately tracking income.

Overtime Hours

Overtime hours are the hours worked beyond the regular hours defined by an employer's policy or the country's labor law. In this situation, the woman worked more than her regular 36-hour workweek. For all these additional hours, she is compensated at a higher rate - \(\\(8.94\) per hour.

To determine how many overtime hours she worked, you need to divide her overtime earnings by the overtime rate. This calculation will show how many extra hours she contributed:

• Overtime Pay Earned = \(\\)71.52\)
• Overtime Hourly Rate = \(\$8.94\)
• Overtime Hours Worked = \(\frac{71.52}{8.94} \approx 8\)

The understanding of overtime is important, as it impacts total earnings. It often offers employees a chance to earn more, as any work beyond standard hours is usually paid at a higher rate. These calculations highlight her effort and the reward for working beyond the usual schedule.