$$ R=\frac{L}{L+D} $$
A streaming website uses the formula above to calculate the rating, \(R\), of its films, based on the number of likes, \(L\), and dislikes, \(D\). Which of the following expresses the number of likes in terms of the other variables?
Though this may seem difficult at first, it becomes trivial if we compare it to a similar case you've probably done before:
Rating
$$ R=\frac{L}{L+D} $$
$$ R(L+D)=L $$
$$ RL+RD=L $$
$$ RL-L=-RD $$
$$ (R-1)L=-RD $$
$$ L=-\frac{RD}{R-1} $$
Solve for x
$$ 2=\frac{x}{x+1} $$
$$ 2(x+1)=x $$
$$ 2x+2(1)=x $$
$$ 2x-x=-2(1) $$
$$ (2-1)x=-2(1) $$
$$ x=-\frac{2(1)}{2-1} $$
We can distribute the negative sign to the denominator to clean up the equation and match one of the options:
$$ L=-\frac{RD}{R-1} $$
$$ L =\frac{RD}{-(R-1)} $$
$$ L=\frac{RD}{-R+1} $$
$$ \boxed{L=\frac{RD}{1-R}} $$
You may be tempted to divide immediately, but we can only do so if we first take the reciprocal of both sides.
$$ R=\frac{L}{L+D} $$
$$ \frac{1}{R}=\frac{L+D}{L} $$
$$ \frac{1}{R}=\frac{L}{L}+\frac{D}{L} $$
$$ \frac{1}{R}=1+\frac{D}{L} $$
$$ \frac{1}{R}-1=\frac{D}{L} $$
$$ \frac{1}{R}-\frac{R}{R}=\frac{D}{L} $$
$$ \frac{1-R}{R}=\frac{D}{L} $$
$$ \frac{R}{1-R}=\frac{L}{D} $$
$$ \boxed{L=\frac{RD}{1-R}}$$