# Thread: Math/algebra help required - practical application (tax)

1. ## Math/algebra help required - practical application (tax)

I've ended up here whilst looking for some help with a calculation, I genuinely can't work out what formulae would be required! As this is basically algebra, I felt this was the most appropriate sub-forum.

The premise for this is calculating employee's gross pay and employers national insurance contributions given that I have a defined amount of money, X, to cover both. Essentially I bill monthly for Y per day, expenses come off the top line and what's left is to cover both my salary and employers NI contributions from the business. Employers NI is not a straight percentage, it's 13.8% of employee's earnings over £589 per month (roughly).

The math/algebra looks as follows;

X = A + B
B = 0.138*(A-589) = X - A
A= (B/0.138)+589 = X - B

How can you work out A and B when you input X (i.e A and B are unknowns). I assume this would be perhaps some iterative process?

Any input appreciated. I can work these out by trial and error with online tax calculators, or by using a third party umbrella company website, but I want to be able to build my own spreadsheet that tells me everything I need to know based on inputting just the number of days I'm billing for.

2. ## Re: Math/algebra help required - practical application (tax)

Actually don't worry, it's basic substitution!

Now, I just need to work out how to get that into excel!

3. ## Re: Math/algebra help required - practical application (tax)

I'm not understanding the difficulty.

You have all the equations. If you know X (input Y and the expenses, then Y-expenses = X), you can find both A and B. You have three equations, and three unknowns.

X=A+B. Knowing X, find A, you can find B and vice versa.

B = X-A= 0.138*(A-589) Solve this for A in terms of X.

X-A = .138A - 589(.138) Add A to both sides, and add 589(.138) to both sides.

X + 589(.138) = 1.138A Divide both sides by 1.138.

(X+589(.138))/1.138=A Now you have A in terms of X. Once you find A, you know B = X-A.

In excel, whatever cell X (call it X1) is in the formula is ...

=(X1+(589*.138))/1.138 That's the A. Say that's in Cell A2

B is

=X1-A2

If Y were 10,000 and expenses were 5,000, then X = 5000.

(5000+589(.138))/1.138 = 4465.0941827768...

That makes B = 534.901581

I don't know if you have to round the tax payment to the hundredth, thousandth, or what decimal (talk to a tax attorney), but by general rounding rules that's 4465.10 paid to employees with 534.90 as the NI.

4. ## Re: Math/algebra help required - practical application (tax)

Thanks for that!

The difficulty was, high school algebra was quite some time ago! Once I remembered substitution I was good to go! Some rearranging later and it's in excel nicely too.