Often there are 2 variables that are in some way connected to each other. This is known as proportionality and there are various methods we can use to investigate how they behave with each other.
direct proportionality is when one variable is directly linked to another variable. In general, we can describe it as:
$$ y \propto x \Rightarrow y = kx$$
We can turn the proportional sign into an equal to sign by adding another letter which is a constant (where the value is a fixed one but unknown) whilst y and x are variables.
Questions about this will usually ask us to find what the value for k (constant) is and from that work out other values after being given what y and x equal at a particular point. For example, “y and x are directly proportional. Given that when y is 5, x is 10 find an equation for the relationship:”
$$ 5 \propto 10 \Rightarrow \dfrac{5}{10} = \dfrac{10k}{10} $$
$$ k = \dfrac{1}{2} $$
After we have found the constant they may ask us to work out work x is equal to when y is a certain value or something similar. Using the above question, let’s work out the value for x when y is 10:
$$ y = kx \Rightarrow 10 \color{red}{\times 2} = 0.5x \color{red}{\times 2}$$ $$ 20 = x $$
Another type of proportionality is known as inversely proportional. In general proportional means when 2 variables are connected in some way, and “inversely” tends to mean 1 divided by the variable. Here is the general form:
$$ y \propto \dfrac{1}{x} \Rightarrow y = \dfrac{k}{x} $$ $$ y = \dfrac{1}{x} \times \dfrac{k}{1} $$
Let’s do a quick inversely proportional question. “y is inversely proportional to the square root of x. When y is 5 x is 36. Work out the equation that link’s these variables together and then work out the value when y is 30:”
$$ y \propto \dfrac{1}{\sqrt{x}} \Rightarrow 5 = \dfrac{k}{\sqrt{36}} $$
$$ 5 \color{red}{\times 6} = \dfrac{k}{\sqrt{36}} \color{red}{\times 6} $$
$$ 30 = k \Rightarrow y = \dfrac{30}{\sqrt{x}} $$
$$ 30 \color{red}{\times \sqrt{x}} = \dfrac{30}{\sqrt{x}} \color{red}{\times \sqrt{x}} $$
$$ \dfrac{30 \sqrt{x}}{\color{red}{30}} = \dfrac{30}{\color{red}{30}} $$
$$ (\sqrt{x})^\color{red}{2} = 1^\color{red}{2} $$ $$ x = 1 $$
For many of you I hope you realised you could solve this intuitively by realising that 30 divided by 1 will give 30 therefore the square root of x must be 1.
More coming soon