Splitting money to equal values after different tax rates

Question

Sorry if the format of this question is slightly off I’m not a regular user but need help. I have Money of £704000. This needs to be split between two people A and B equally after tax A has tax rate of 11.75% B has tax rate of 23.5% How would I go about solving this?

Answer 1

Let's build this from the ground up. We want to split an amount of $704~000$ to A and B so that after their taxes have been deducted, they both get the same amount $x$.

Let's first imagine that we split the amount into two boxes, without deducting any taxes. $$\tag{1} 704~000 = A + B $$ $A$ is the part that will go to A and $B$ is the part that will go to $B$. Now let's think about the taxes. If A has the amount $A$ before the taxes, how much is there after taxes? We know that A's tax percentage is $11.75~\%$. That means that after the deduction, A will have $$ \underbrace{A}_{\text{original amount}} - \underbrace{0.1175 A}_{\text{taxes}} = A(1-0.1175) $$ So this is what A will get after the taxes and this should be equal to $x$. So we know $$ A(1-0.1175) = x \qquad \Rightarrow \qquad A = \frac{x}{1-0.1175} $$ This was pure algebra.

Next, we do the same thing for B. He also has to to pay taxes, so the amount after taxes is $$ B - 0.235B = B(1-0.235) = x \qquad \Rightarrow \qquad B = \frac{x}{1-0.235} $$ Now we have alternative expressions for $A$ and $B$, and we can plug them into Equation ($1$): $$ \begin{split} 704~000 &= \frac{x}{1-0.1175} + \frac{x}{1-0.235}\\ &= \left( \frac{1}{1-0.1175} + \frac{1}{1-0.235} \right) x \\ &\approx (1.1331 + 1.3072)x \\ & = 2.4403 x \end{split} $$ Dividing both sides by $2.4403$, we get $$ x = \frac{704~000}{2.4403} \approx 288~485.1 $$

Answer: both get approximately $288~485.1$ pounds.