If I had a mixture of $A$ and $B$, and I wanted to increase $A$ so that $B$ drops to $90\%$ of what it was in the solution -- how would I solve this?
For example, say $A$ was $90$ grams ($90\%$) and $B$ was $10$ grams ($10\%$) for a total of $100$ grams. By how much I increase the weight of $A$ so that $B$ becomes $9\%$?
If B is $10$% of the mixture, we can say that the ratio of $\frac{B}{A+B} =0.1$
Relative to your example, to solve for the increase in A such that B becomes $9$ percent of the solution, whilst the amount of $B$ remains at $10$ grams we input:
$\frac{B}{A+B} = 0.09$
$\frac{10}{A+10} = 0.09$
$A = \frac{10}{0.09} - 10$
$A = \frac{1000}{9} - \frac{90}{9}$
$A = \frac{910}{9}$
Furthermore, if we want to solve for the change in A we can compute:
$\Delta A = A_{final} - A_{initial}$
$\Delta A = \frac{910}{9} - 90$
$\Delta A = \frac{100}{9}$