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Let's go through the problems one by one:

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**Problem 12**:  
The electric current \( I \) (in amperes) varies directly as the voltage \( V \). When \( 18 \) volts are applied, the current is \( 4 \) amperes. What is the current when \( 24 \) volts are applied?

We are dealing with direct proportionality, so we can write the relationship as:

\[
I = kV
\]

where \( k \) is the constant of proportionality. Using the given data \( I = 4 \) when \( V = 18 \), we can solve for \( k \):

\[
4 = k \cdot 18 \quad \Rightarrow \quad k = \frac{4}{18} = \frac{2}{9}
\]

Now, to find the current when \( V = 24 \):

\[
I = \frac{2}{9} \cdot 24 = \frac{48}{9} = 5.33 \text{ amperes}
\]

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**Problem 13**:  
Casey’s weekly pay is directly proportional to the number of hours he works at the YMCA. His gross earnings are \( 210 \) dollars for \( 24 \) hours. What is Casey’s pay if he works \( 40 \) hours?

This is also a direct proportionality problem, so we can set up the equation:

\[
P = kH
\]

where \( P \) is pay, \( H \) is hours worked, and \( k \) is the constant of proportionality. Using \( P = 210 \) and \( H = 24 \):

\[
210 = k \cdot 24 \quad \Rightarrow \quad k = \frac{210}{24} = 8.75
\]

Now, to find Casey's pay for \( 40 \) hours:

\[
P = 8.75 \cdot 40 = 350 \text{ dollars}
\]

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**Problem 14**:  
The weight \( M \) of an object on Mars is directly proportional to its weight \( E \) on Earth. An astronaut who weighs \( 170 \) lbs on Earth weighs \( 64 \) lbs on Mars. How much would a \( 125 \) lb astronaut weigh on Mars?

We use the direct proportionality again:

\[
M = kE
\]

Using the given data \( M = 64 \) when \( E = 170 \):

\[
64 = k \cdot 170 \quad \Rightarrow \quad k = \frac{64}{170} = 0.376
\]

Now, to find the weight on Mars for a \( 125 \) lb astronaut:

\[
M = 0.376 \cdot 125 = 47 \text{ lbs}
\]

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Would you like any details on the steps above or have any questions?

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Here are 5 additional questions to think about:

1. How would you solve Problem 12 if the voltage was tripled?
2. What happens to the pay in Problem 13 if Casey only works 10 hours?
3. In Problem 14, what would happen if an astronaut weighed 200 lbs on Earth?
4. How would you find the constant \( k \) if the relationships were not direct but inverse proportionality?
5. What units would the constant \( k \) have in the equation for Problem 12?

**Tip**: When solving proportionality problems, always isolate the constant first and then apply it to new values.

Solve the three direct proportionality problems involving current and voltage, weekly pay, and weight on Mars.

Solve problems involving direct proportionality: electric current and voltage, pay and hours worked, and weight on Mars compared to Earth. Detailed step-by-step solutions provided for students in Grades 8-10.

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