I am hoping that either someone else can point me in the right direction and or I step away from this problem long enough to figure out the solution. Either way here is the problem I am trying to solve.
Basically, I have a numerator and denominator which as an end result equals a percentage. That percentage will never exceed 100%. The numerator in this example is the number of red gems and the denominator is a total of both red and blue gems. The target percentage of red gems compared to the total gems should not exceed 70%. However, as an example say I have a percentage of 45% how many more red gems would I need to proportionally reach as close as possible to 70%?
$$ \frac{45+n}{100+n}<=.70 $$
Thanks in advance
That of course depends on how many total gems, or how many red gems, you have.
If $g$ is the total number of gems you have and $45\%$ of your gems are red, that means you have $0.45g$ red gems. If you then add $n$ red gems to your collection, you will now have $g+n$ total gems, and $0.45g+n$ red gems. You want to solve, for $n$, the equality
$$\frac{0.45g+n}{g+n}=0.7$$
This is equivalent to
\begin{align} 0.45g+n=0.7\times (g+n) &\iff 0.45g+n=0.7g+0.7n\\ &\iff 0.3n=0.25g\\ &\iff n=\frac{0.25g}{0.3}=\frac{5}{6}g \end{align}
In other words, you need to add red gems equal to $83\%$ of your current total gem collection.