Sharon earns a base pay of $14/h. She also earns time and a half for overtime. Overtime is any hours beyond 40 h. The goal of the question is to write an equation to model the situation. Then use the equation to determine how many hours Sharon worked if she earned $749.

Question

Complete the equation that models Sharon’s earnings, E, where t is the number of overtime hours. Assume she works at least 40 h. (Hint: Think of her earnings as the amount earned for 40 h plus the amount earned for any overtime hours.)

E = __t + ___

Answer 1

To model Sharon’s earnings, we first need to consider how her earnings break down.

1. Base Pay Calculation: Sharon has a base pay of $14/hour for the first 40 hours of work. For 40 hours, her earnings can be calculated as: \[ \text{Base earnings for 40 hours} = 14 \times 40 = 560 \]

2. Overtime Pay Calculation: She earns time and a half for any overtime hours (beyond 40 hours). Time and a half means her overtime pay rate is: \[ \text{Overtime rate} = 14 \times 1.5 = 21 \] If \( t \) is the number of overtime hours, then her earnings from overtime can be calculated as: \[ \text{Earnings from overtime} = 21t \]

3. Total Earnings Equation: The total earnings \( E \) can be expressed as the sum of her earnings for the first 40 hours plus her overtime earnings: \[ E = 21t + 560 \]

So, the complete equation modeling Sharon's earnings is: \[ E = 21t + 560 \]

Next, we can use this equation to determine how many hours Sharon worked if she earned $749.

Setting \( E = 749 \): \[ 749 = 21t + 560 \]

Now, we can solve for \( t \):

1. Subtract 560 from both sides: \[ 749 - 560 = 21t \] \[ 189 = 21t \]

2. Divide by 21: \[ t = \frac{189}{21} = 9 \]

This means Sharon worked 9 hours of overtime. Since she works at least 40 hours, the total hours she worked is: \[ \text{Total hours} = 40 + t = 40 + 9 = 49 \]

In summary, Sharon worked a total of 49 hours.