Question

1. diego worked out a deal with his parents. for every hour that he reads a book, he earns \\(\frac{1}{4}\\)-hour of screen time. diego uses the equation \\(s = \frac{1}{4}r\\) to represent this relationship.

a. what does each variable in the equation represent?
b. which is the independent variable? which is the dependent variable? explain how you know.

Explanation:

Part a

Step1: Analyze the equation

The equation given is \( s=\frac{1}{4}r \). Here, we need to determine what each variable represents. From the problem context, Diego earns \(\frac{1}{4}\)-hour of screen time for every hour he reads a book. Let's define the variables:

• Let \( r \) represent the number of hours Diego reads a book (the independent quantity, as it's the input or the quantity we can choose freely).
• Let \( s \) represent the number of hours of screen time Diego earns (the dependent quantity, as it depends on the number of hours he reads).

So, in the equation \( s = \frac{1}{4}r \), \( r \) is the number of hours spent reading a book, and \( s \) is the number of hours of screen time earned.

Part b

Step1: Recall variable definitions

In a relationship between two variables, the independent variable is the one that is manipulated or chosen freely, and the dependent variable is the one that depends on the independent variable.

• Independent Variable: \( r \) (the number of hours Diego reads a book). This is because the amount of reading time is a quantity that Diego (or the situation) can control or vary, and the screen time depends on it.
• Dependent Variable: \( s \) (the number of hours of screen time earned). This is because the screen time "depends" on the number of hours spent reading. If the number of reading hours (\( r \)) changes, the screen time (\( s \)) will change according to the equation \( s=\frac{1}{4}r \). For example, if \( r = 4 \) hours (reading), then \( s=\frac{1}{4}\times4 = 1 \) hour (screen time); if \( r = 8 \) hours, then \( s=\frac{1}{4}\times8 = 2 \) hours. So \( s \) depends on \( r \).

Part a Answer:

In the equation \( s=\frac{1}{4}r \), \( r \) represents the number of hours Diego reads a book, and \( s \) represents the number of hours of screen time Diego earns.

Part b Answer:

• Independent variable: \( r \) (number of hours reading a book). It is independent because the amount of reading time can be chosen or varied freely, and the screen time depends on it.
• Dependent variable: \( s \) (number of hours of screen time earned). It is dependent because its value is determined by the number of hours spent reading (the value of \( r \)) according to the equation \( s = \frac{1}{4}r \).

Answer:

Step1: Recall variable definitions

In a relationship between two variables, the independent variable is the one that is manipulated or chosen freely, and the dependent variable is the one that depends on the independent variable.

• Independent Variable: \( r \) (the number of hours Diego reads a book). This is because the amount of reading time is a quantity that Diego (or the situation) can control or vary, and the screen time depends on it.
• Dependent Variable: \( s \) (the number of hours of screen time earned). This is because the screen time "depends" on the number of hours spent reading. If the number of reading hours (\( r \)) changes, the screen time (\( s \)) will change according to the equation \( s=\frac{1}{4}r \). For example, if \( r = 4 \) hours (reading), then \( s=\frac{1}{4}\times4 = 1 \) hour (screen time); if \( r = 8 \) hours, then \( s=\frac{1}{4}\times8 = 2 \) hours. So \( s \) depends on \( r \).

Part a Answer:

In the equation \( s=\frac{1}{4}r \), \( r \) represents the number of hours Diego reads a book, and \( s \) represents the number of hours of screen time Diego earns.

Part b Answer:

• Independent variable: \( r \) (number of hours reading a book). It is independent because the amount of reading time can be chosen or varied freely, and the screen time depends on it.
• Dependent variable: \( s \) (number of hours of screen time earned). It is dependent because its value is determined by the number of hours spent reading (the value of \( r \)) according to the equation \( s = \frac{1}{4}r \).