Directly proportional quantities
Two quantities $A$ and $B$ are directly proportional to each other if their ratio is a constant; in formulas:
$$frac{A}{B}=k$$
$k$ is called the constant of direct proportionality.
Example
Let’s consider the weight of a certain quantity of oranges ($A$) and its price per kg ($B$).
| Weight of oranges (Kg) | Price per Kg (€) |
| $1$ | $2$ |
| $2$ | $4$ |
| $3$ | $6$ |
| $4$ | $8$ |
We note that as the weight increases, the price also increases and furthermore, the ratio between price and weight always remains constant:
$$frac{2}{1}=frac{4}{2}=frac{6}{3}=frac{8}{4}=2$$
We can write that
$$frac{price}{weight}=2$$
that is, the constant of direct proportionality is $2$.
Inversely proportional quantities
Two quantities $A$ and $B$ are inversely proportional to each other if their product is a constant; in formulas:
$$Acdot B=k$$
$k$ is called the constant of inverse proportionality.
Example
Let us consider the number of sides of a regular polygon with perimeter $360cm$ ($A$) and the length of its side expressed in cm ($B$).
| Number of sides | Side measurement (cm) |
| $3$ | $120$ |
| $4$ | $90$ |
| $5$ | $72$ |
| $6$ | $60$ |
We note that as the number of sides increases, the size of the single side decreases and furthermore, the product of the two quantities always remains constant:
$$3*120=4*90=5*72=6*60=360$$
We can write that
$$n° sides*side size=360$$
that is, the constant of inverse proportionality is $360$.
Example
Given the following table showing some values of the pairs $x$ and $y$. Identify the type of proportionality and the constant of proportionality.
| x | y |
| $1$ | $3$ |
| $2$ | $6$ |
| $3$ | $9$ |
| $4$ | $12$ |
First, we note that as $x$ increases, $y$ also increases. Furthermore, their ratio always remains constant:
$$frac{3}{1}=frac{6}{2}=frac{9}{3}=frac{12}{4}=3$$
We have that $y$ is directly proportional to $x$ with a direct proportionality constant equal to $3$.
Example
Given the following table showing some values of the pairs $x$ and $y$. Identify the type of proportionality and the constant of proportionality.
| x | y |
| $80$ | $40$ |
| $50$ | $64$ |
| $25$ | $128$ |
| $10$ | $320$ |
First, we note that as $x$ decreases, $y$ increases. Furthermore, their product always remains constant:
$$80*40=50*64=25*128=10*320=3200$$
We have that $y$ is inversely proportional to $x$ with an inverse proportionality constant equal to $3200$.
Exercises to do
| x | y |
| $7$ | $1$ |
| $14$ | $2$ |
| $28$ | $4$ |
| $35$ | $5$ |
| x | y |
| $3$ | $6$ |
| $9$ | $2$ |
| $18$ | $1$ |
| $27$ | $frac{2}{3}$ |
1) Given the following table showing some values of the pairs $x$ and $y$. Identify the type of proportionality and the constant of proportionality.
2) Given the following table showing some values of the pairs $x$ and $y$. Identify the type of proportionality and the constant of proportionality.
