I have a total amount excl VAT of 100, with 2 parts: 1 = 90% of the total, 1 = 10% of the total. Both parts have a VAT rate which need to be added for part 1 = 21% and part 2 = 6%. The total amount incl. VAT would be 119,5.
Can you now calculate back to the total amount excl tax when you have the total amount incl tax (119,5) , the percentages of the parts (90% and 10 %) and the VAT percentages of the parts (21% and 6%)?
First we create the following variables:
\begin{align}t&=\text{total before VAT}\\ T&=\text{total after VAT}\\ p_1&=\text{percentage for part 1}\\ p_2&=\text{percentage for part 2}\\ v_1&=1+\text{VAT rate for part 1}\\ v_2&=1+\text{VAT rate for part 2}\end{align}
Note that, for example, if our VAT rate was listed as $28\%$ then $v_1$ will be $1.28$ - this is the number we need to multiply the before VAT value by to get the after VAT value
Now we can say that
$$T = p_1tv_1+p_2tv_2$$
We can then input the numbers that we know to get $$119.5=0.9t\times1.21+0.1t\times 1.06$$
We then simply solve this equation for $t$ as follows:
\begin{align}119.5&=0.9t\times1.21+0.1t\times 1.06\\ 119.5&=1.089t+0.106t\\ 119.5&=1.195t\\ t&=\frac{119.5}{1.195}\\ &=100\end{align}
We can also note that\begin{align}T &= p_1tv_1+p_2tv_2\\ T&=t(p_1v_1+p_2v_2)\\ t&=\frac T{p_1v_1+p_2v_2}\end{align}