How would I find the number of people who rated a product by knowing that everyone rated $5$ stars and when I entered a rating of $1$ the average rating went down to $4.5$.
Is there a way to find out how many people rated? I could do it manually adding $1+5+5$ and so on and divide by how many numbers are there, but is there a formula to do it?
Let the number of people who rated $5$ stars be $x$.
Then, we know that the total number of stars in the beginning is $5x$ and the total number of stars in the end is $4.5(x+1)$. Hence, we have the equality
$$ 5x+1=4.5(x+1)=4.5x+4.5 \\ 0.5x=3.5 \\ x=7 $$
Hence, the number of people who rated is $\boxed{7}$.
Generalisation
Suppose that $x$ people has rated and the average is $y$. Now, a new person votes $z$ and the average went down to $w$.
Then, we have the equality
$$ xy+z=w(x+1)=wx+w \\ x(y-w)=w-z \\ x=\frac{w-z}{y-w} $$
Hence, the general formula for $x$ is $\boxed{\dfrac{w-z}{y-w}}$.