Since the paycheck, \( p \), varies directly with the number of hours worked, \( t \), we can express this relationship with the equation:
\[ p = kt \]
where \( k \) is the constant of proportionality.
To find the value of \( k \), we can use the information given in the problem. We know that for 25 hours of work, the paycheck is $312.50. Substituting these values into the equation gives:
\[ 312.50 = k \times 25 \]
Now, we can solve for \( k \):
\[ k = \frac{312.50}{25} = 12.50 \]
Now that we have found \( k \), we can substitute it back into the original equation to get the final equation that relates hours worked to pay:
\[ p = 12.50t \]
This equation means that for every hour worked, the person earns $12.50. This is how we derived the equation for the relationship between hours worked and pay.