Understanding the Relationship Between Hours Worked and Total Pay

Oct 13, 2025 by miodeya.com 23 views

Understanding the Relationship Between Hours Worked and Total Pay

When it comes to calculating total pay, it's essential to understand the relationship between the number of hours worked and the amount earned. In this scenario, Noah, a paralegal, received a $\$500$ signing bonus and makes $\$28$ an hour. To model his total pay $y$ in dollars as it relates to the number of hours $x$ that he works, we need to create an equation.

The Equation: A Linear Relationship

The equation that models Noah's total pay $y$ in dollars as it relates to the number of hours $x$ that he works is a linear equation. A linear equation is a polynomial equation of degree one, which means it has the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

In this case, the slope $m$ represents the rate at which Noah's pay increases per hour worked. Since he makes $\$28$ an hour, the slope is $28$ . The y-intercept $b$ represents the initial amount that Noah earned, which is the $\$500$ signing bonus.

The Equation: $y = 28x + 500$

Therefore, the equation that models Noah's total pay $y$ in dollars as it relates to the number of hours $x$ that he works is:

y = 28x + 500

This equation indicates that for every hour Noah works, his pay increases by $\$28$ . Additionally, the $\$500$ signing bonus is added to his total pay.

Interpreting the Equation

To understand the equation better, let's break it down:

The slope $m = 28$ represents the rate at which Noah's pay increases per hour worked.
The y-intercept $b = 500$ represents the initial amount that Noah earned, which is the $\$500$ signing bonus.
The equation $y = 28x + 500$ indicates that for every hour Noah works, his pay increases by $\$28$ .

Example: Calculating Total Pay

Let's use the equation to calculate Noah's total pay for working 10 hours.

y = 28x + 500

Substitute $x = 10$ into the equation:

y = 28(10) + 500

y = 280 + 500

y = 780

Therefore, Noah's total pay for working 10 hours is $\$780$ .

Conclusion

In conclusion, the equation $y = 28x + 500$ models Noah's total pay $y$ in dollars as it relates to the number of hours $x$ that he works. This equation indicates that for every hour Noah works, his pay increases by $\$28$ , and the $\$500$ signing bonus is added to his total pay. By understanding this equation, we can calculate Noah's total pay for any number of hours worked.

Understanding the Graph

The graph of the equation $y = 28x + 500$ is a straight line with a slope of 28 and a y-intercept of 500. The graph can be used to visualize the relationship between the number of hours worked and the total pay.

The Graph: A Straight Line

The graph of the equation $y = 28x + 500$ is a straight line that passes through the points (0, 500) and (10, 780). The graph indicates that for every hour Noah works, his pay increases by $\$28$ .

Interpreting the Graph

To understand the graph better, let's break it down:

The slope of the graph represents the rate at which Noah's pay increases per hour worked.
The y-intercept of the graph represents the initial amount that Noah earned, which is the $\$500$ signing bonus.
The graph indicates that for every hour Noah works, his pay increases by $\$28$ .

Example: Using the Graph

Let's use the graph to calculate Noah's total pay for working 10 hours.

Find the point on the graph that corresponds to 10 hours worked.
The point on the graph is (10, 780).
Therefore, Noah's total pay for working 10 hours is $\$780$ .

Conclusion

In conclusion, the graph of the equation $y = 28x + 500$ is a straight line that represents the relationship between the number of hours worked and the total pay. The graph can be used to visualize the relationship and calculate the total pay for any number of hours worked.

Real-World Applications

The equation $y = 28x + 500$ has real-world applications in various fields, such as:

Business: The equation can be used to calculate the total pay for employees who receive a signing bonus and hourly wages.
Finance: The equation can be used to calculate the total interest paid on a loan or investment.
Science: The equation can be used to model the relationship between variables in scientific experiments.

Conclusion

In the previous article, we discussed the equation $y = 28x + 500$ and how it models Noah's total pay $y$ in dollars as it relates to the number of hours $x$ that he works. In this article, we will answer some frequently asked questions about the equation and provide additional insights.

Q: What is the slope of the equation $y = 28x + 500$ ?

A: The slope of the equation $y = 28x + 500$ is 28. This represents the rate at which Noah's pay increases per hour worked.

Q: What is the y-intercept of the equation $y = 28x + 500$ ?

A: The y-intercept of the equation $y = 28x + 500$ is 500. This represents the initial amount that Noah earned, which is the $\$500$ signing bonus.

Q: How can I use the equation $y = 28x + 500$ to calculate Noah's total pay?

A: To calculate Noah's total pay, simply substitute the number of hours worked into the equation. For example, if Noah works 10 hours, the equation becomes:

y = 28(10) + 500

y = 280 + 500

y = 780

Therefore, Noah's total pay for working 10 hours is $\$780$ .

Q: What if I want to calculate Noah's total pay for working a fraction of an hour?

A: To calculate Noah's total pay for working a fraction of an hour, simply substitute the fraction of an hour into the equation. For example, if Noah works 3.5 hours, the equation becomes:

y = 28(3.5) + 500

y = 98 + 500

y = 598

Therefore, Noah's total pay for working 3.5 hours is $\$598$ .

Q: Can I use the equation $y = 28x + 500$ to calculate the number of hours worked?

A: Yes, you can use the equation $y = 28x + 500$ to calculate the number of hours worked. To do this, simply rearrange the equation to solve for $x$ . The resulting equation is:

x = \frac{y - 500}{28}

For example, if Noah's total pay is $\$780$ , the equation becomes:

x = \frac{780 - 500}{28}

x = \frac{280}{28}

x = 10

Therefore, Noah worked 10 hours.

Q: What if I want to calculate the number of hours worked for a fraction of an hour?

A: To calculate the number of hours worked for a fraction of an hour, simply substitute the fraction of an hour into the equation. For example, if Noah's total pay is $\$598$ , the equation becomes:

x = \frac{598 - 500}{28}

x = \frac{98}{28}

x = 3.5

Therefore, Noah worked 3.5 hours.

Q: Can I use the equation $y = 28x + 500$ to compare the pay of different employees?

A: Yes, you can use the equation $y = 28x + 500$ to compare the pay of different employees. To do this, simply substitute the number of hours worked and the hourly wage into the equation. For example, if two employees work the same number of hours but have different hourly wages, the equation can be used to compare their total pay.

Q: What if I want to calculate the total pay for an employee who receives a signing bonus and hourly wages, but also has a commission-based pay?

A: To calculate the total pay for an employee who receives a signing bonus and hourly wages, but also has a commission-based pay, you will need to use a more complex equation that takes into account the commission-based pay. The equation will be in the form:

y = 28x + 500 + cx

Where $c$ is the commission rate.

Conclusion

In conclusion, the equation $y = 28x + 500$ is a powerful tool for calculating total pay and understanding the relationship between hours worked and pay. By answering these frequently asked questions, we hope to have provided additional insights and clarity on how to use the equation in real-world applications.