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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?

Henry Cooper
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  • So what you mean is that in the end (after taxes), both A and B receive the same amount, right? In that case, let us call that amount $x$. How much is the amount for $A$ before taxes? That's of course $$ \frac{x}{1-\frac{11.75}{100}} $$ This is because this number multiplied by $\left( 1- \frac{11.75}{100} \right)$ must be equal to $x$.The same figure for $B$ is $$ \frac{x}{1- \frac{23.5}{100}} $$ The sum of these two numbers is then $704~000$. – Matti P. Oct 17 '19 at 11:23
  • So how would I go about solving. You have essentially 0.8825 and 0.765 which is the money they keep. How do I then spilt up 704000 so they each keep equal amounts?? – Henry Cooper Oct 17 '19 at 11:31
  • So you get the equation $$ \frac{x}{1-\frac{11.75}{100}} + \frac{x}{1-\frac{23.5}{100}} = 704~000 $$ Can you solve it? It's a linear equation... – Matti P. Oct 17 '19 at 11:32
  • Sorry it’s been a long time since I did actual maths. I’m having trouble. – Henry Cooper Oct 17 '19 at 11:40

1 Answers1

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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.

Matti P.
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