Differential equation word problem

There are 500 students enrolled for a certain course. A rumour starts to spread

among these students that the next class test has been cancelled. Suppose that the rate at which this rumour spreads is proportional to the product of the fraction of the students in the course who have heard the rumour and the fraction of students in the course who have not yet heard it. At 8:00 am, 40 students have heard the rumour, and by 12:00 noon half the students have heard it. By what time would 90% of the students have heard the rumour?

I checked the answer and I didnt understand why the derivative used was:

$\displaystyle \frac{dp}{dt} = kp(1 - p)$

p = % of students who have heard the rumour

t = time at which these % of students hear the rumour

Which part on the right-hand side of this equation refers to:

*product of the fraction of the students in the course who have heard the rumour?

*fraction of students in the course who have not yet heard it?