Direct and inverse proportion

Example

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:

$$\begin{array}{rcl}3\text{h}& \to & 400\text{km}\\ x\text{h}& \to & 800\text{km}\end{array}$$

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:

$${\displaystyle \frac{3}{x}}={\displaystyle \frac{400}{800}}$$

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:

$$x={\displaystyle \frac{800\cdot 3}{400}}={\displaystyle \frac{2400}{400}}=6$$

Therefore it will take $6$ hours for the train to do $800$ km.

Example

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:

$$\begin{array}{rcl}1\text{kg}& \to & 4,5\text{€}\\ \frac{1}{2}\text{kg}& \to & x\text{€}\end{array}$$

Applying the rule of three we have:

$$x={\displaystyle \frac{\frac{1}{2}\cdot 4,5}{1}}={\displaystyle \frac{1}{2}}\cdot 4,5=2,25$$ €

So, half a kilo of cherries will cost half the price of one kilo.

Example

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:

$$\begin{array}{rcl}2\text{farmers}& \to & 10\text{days}\\ 5\text{farmers}& \to & x\text{days}\end{array}$$

And it is solved:

$$x={\displaystyle \frac{2\cdot 10}{5}}={\displaystyle \frac{20}{5}}=4$$ 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.