"In mathematics, a rate is the ratio between two related quantities in different units. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable."
Can someone explain this with an example? Say using S = vt
On
For $s=vt$, this can be rearranged to $$ v=\frac{s}{t} \, . $$ In other words, your velocity is equal to your displacement divided by time. Here, time is an independent variable—you are measuring your velocity against the amount of time that has been passed. This notion of 'independent variable' becomes even more important in calculus, where the above equation becomes $$ v=\frac{ds}{dt} \, . $$ Here, $dt$ can (roughly speaking) be interpreted as infinitesimal change in time. Then, the 'ratio' $$ \frac{ds}{dt} $$ an infinitesimal change in displacement divided by the infinitesimal change in time.
Beware: the $d$'s in the notation $$ \frac{ds}{dt} $$ cannot be cancelled. They just represent infinitesimal change. So $$ dt = \text{infinitely small change in time.} $$
Comment
Some examples
Speed, Price of a commodity are constants and they can be expressed as a constant ratio of a single dependent variable in terms and a single independent variable:
$$ \text{ 60 miles/hour, 5 dollars/pound };$$
If the variation is with respect to two independent variables like Interest rate, Acceleration respectively:
$$ \text{ 4 dollars /100 dollars /annum; 25 meters/sec^2};$$