A proportion is just an equality between two or more fractions:
$$$\dfrac{a}{b}=\dfrac{c}{d}$$$
where $$a$$ and $$d$$ are named ends, and $$b$$ and $$c$$, middles.
Direct proportion
We will say that the proportion is direct if they relate magnitudes in which, when increasing one the other also increases, and vice versa.
In this case the rule of three will be applied in the following way:
If it takes $$3$$ hours for a train to travel $$400$$ kilometers: how long will it take to cover double the distance?
To begin, we observe that it is a case of direct proportion since the more hours we spend the more kilometers the train will run. The answer can be deduced mentally since if the train has to do twice the distance, it will also take twice the time, so it will need $$6$$ h to do $$800$$ km. The deduction is correct, but let's see how it is solved applying the rule of three for direct proportions.
We have the following relation:
$$$\begin{eqnarray} 3 \ \mbox{h} & \rightarrow & 400 \ \mbox{km}\\ x \ \mbox{h} & \rightarrow & 800 \ \mbox{km} \end{eqnarray}$$$
That is to say, if in $$3$$ h $$400$$ km are done, in $$x$$ h $$800$$ will be done.
We observe that the relation can also be expressed following the equality model between fractions used to describe the concept of proportion:
$$$\dfrac{3}{x}=\dfrac{400}{800}$$$
Where both magnitudes of the exercise stay in different fractions: the time in one side of the equality and the distance in the other.
Now we only have to isolate the $$x$$ to find the solution:
$$$x=\dfrac{800\cdot3}{400}=\dfrac{2400}{400}=6$$$
Therefore it will take $$6$$ hours for the train to do $$800$$ km.
If a kilo of cherries costs $$4,5$$ €: how much will half a kilo cost?
We have a direct proportionality since the less kilos we buy the less they will cost.
We have the proportionality relation:
$$$\begin{eqnarray} 1 \ \mbox{kg} & \rightarrow & 4,5 \ \mbox{€}\\ \frac{1}{2} \ \mbox{kg} & \rightarrow & x \ \mbox{€} \end{eqnarray}$$$
Applying the rule of three we have:
$$$x=\dfrac{\frac{1}{2}\cdot4,5}{1}=\dfrac{1}{2}\cdot4,5=2,25$$$ €
So, half a kilo of cherries will cost half the price of one kilo.
Inverse proportion
We will say that the proportion is inverse if it involves a magnitudes relation where, when we increase one of them, the other one decreases, and vice versa. In this case the rule of three will be applied as follows:
If it takes $$10$$ days for $$2$$ farmers to plow a field: how much will it take the same work for $$5$$ farmers?
This is a clear example of inverse proportion since the more farmers working the less time will it take to plow the same field.
To solve it the rule of three is applied as taught:
$$$\begin{eqnarray} 2 \ \mbox{farmers} & \rightarrow & 10 \ \mbox{days}\\ 5 \ \mbox{farmers} & \rightarrow & x \ \mbox{days} \end{eqnarray}$$$
And it is solved:
$$$x=\dfrac{2\cdot10}{5}=\dfrac{20}{5}=4$$$ days
So, while it takes $$10$$ days for two farmers, with the help of another $$3$$ partners they'd manage to do the same work in only $$4$$ days.