I have a task, and it's children. Now each child is having an amount and a estimate, which is a said percentage of the amount. So for example, if I have an amount of 100 and an estimate of this amount over 10% is 110.
So I want to calculate the sum of amount of the parent, as well as the estimate, which is quite easy, just add the numbers, for each child.
How do I get an aggregate percentage of the parent though?
eg:
parent1
child1 => 100 - 10% - 110 child2 => 200 - 20% - 240
Now in this case the parent's record will become 300 - x - 350.
I want a formula to find x here. I know it can be found out using 350 and 300, but assuming there are n percentages, how can I calculate the aggregate percentage using all these child percent?
Suppose $a_i$ grows to $a_i(1+r_i),\quad a_i>0,r_i>0, i=1,...,n$.
Then $\sum_{i=1}^n a_i$ grows to $\sum_{i=1}^n a_i(1+R)$ where
$$R:=\frac{\sum_{i=1}^n a_ir_i}{\sum_{i=1}^n a_i},$$
which is a weighted average of the growth rates $r_i$ (weighted by their respective amounts $a_i$). In the special case $r_i=r$ for all $i$, then $R=r.$
In the example you give, $a_1=100,r_1=0.10,a_2=200,r_2=0.20.$ Then the sum $100+200$ grows to $100(1+0.10)+200(1+0.20)=350,$ a relative increase of
$$R=\frac{100\times 0.1+200\times 0.2}{100+200}=0.1\bar{6},$$
as desired: $300(1+0.1\bar{6})=350$.