https://file.aiquickdraw.com/mathbot/file/1728933153112t5ahsswi.jpg

Let's break down and answer the questions in the image.

### (a) How much money does Carl earn per hour?

We can determine Carl's hourly wage by using the data in the table. Let’s take two points from the table, for instance:

\[
\text{For } t = 3 \text{ hours}, \text{ Carl earns } 45 \text{ dollars}.
\]
\[
\text{For } t = 5 \text{ hours}, \text{ Carl earns } 75 \text{ dollars}.
\]

To calculate Carl’s hourly wage, we can compute the rate of change (the slope):

\[
\text{Rate of change} = \frac{\Delta \text{money earned}}{\Delta \text{time}} = \frac{75 - 45}{5 - 3} = \frac{30}{2} = 15 \text{ dollars per hour}.
\]

So, Carl earns **$15 per hour**.

### (b) Who earns more per hour?

We know that Louis’s earnings are described by the equation:

\[
y = 12x,
\]

where $ y $ is the amount of money Louis earns, and $ x $ is the number of hours he works. This means Louis earns **$12 per hour**.

From part (a), we calculated that Carl earns **$15 per hour**.

Thus, **Carl earns more per hour** than Louis, since $15 > $12.

### (c) Draw a graph that represents Carl’s earnings over time.

To plot Carl’s earnings, we use the points from the table:

- At $ t = 3 $ hours, Carl earns $ 45 $ dollars.
- At $ t = 5 $ hours, Carl earns $ 75 $ dollars.
- At $ t = 8 $ hours, Carl earns $ 120 $ dollars.
- At $ t = 10 $ hours, Carl earns $ 150 $ dollars.

The graph should have **time** on the x-axis (in hours) and **money earned** on the y-axis (in dollars). Since Carl earns a constant rate of $15 per hour, the graph will be a straight line with a slope of 15. 

### Would you like me to create the graph for you?

---

**Follow-up Questions:**
1. How do we calculate the rate of change from a table of values?
2. What does the slope of a line represent in real-world problems like this?
3. How would Carl's earnings change if his hourly rate increased to $20 per hour?
4. Can we calculate Louis's total earnings after working 8 hours using his equation?
5. What would Carl's equation for earnings look like, similar to Louis's equation?

**Tip:** When dealing with linear relationships, the slope or rate of change represents how much one quantity changes for every unit increase in the other quantity.

The equation y = 12x describes the amount of money Louis earns, where x is the number of hours he works and y is the amount of money he earns. The table shows the amount of money Carl earns for different numbers of hours worked. (a) How much money does Carl earn per hour? (b) Who earns more per hour? Justify your answer. (c) Draw a graph that represents Carl’s earnings over time in hours.

Solve a real-world problem comparing the hourly wages of Louis and Carl using linear equations. Learn how to calculate the rate of change, determine who earns more, and graph Carl’s earnings over time. Ideal for students in Grades 6-8.

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