Hey all,
So, as the title might imply, I will never receive the McArthur Genius Award on the basis of math abilities. But I’m really trying to figure out the answer to this question.
Let’s say that someone gets $3000 a month, but that income is NOT taxable (federal or state). What’s the best way to figure out what the equivalent of that income would be if it were taxable? Obviously, it would be more than $3000, but that’s where my math abilities kind of break down… all advice appreciated. 
We’d need to know the tax rate, of course. For the sake of example, suppose that you’re in the 20% tax bracket, and represent the amount you’d get before taxes as x. So x - 0.20*x = $3000. Working through the algebra:
x*(1-0.20) = $3000
x*0.80 = $3000
x = $3000 ÷ 0.80
x = $3750
I think I get it! Kind of. or at least, I can follow your math. 
The actual situation is a whole lot more complex than that, but that’s all I need to know for now.
Oh! Wait!! I just KNEW it was more complicated than that… 
The income is not taxable in any way whatsoever. No state or income tax, no Social Security, no Medicare, no nothing. (The explanation for how this is possible is a very long story, so let’s leave it at that.)
So… (tries to think, always difficult with math…) how would you figure this out now?
It doesn’t matter where the % comes from, since the percentiles are additive. All of them apply to the same basis (the pre-tax income).
That 20% could be (20% income tax) or it could be (15% income + 5% SS), it comes to the same thing.
Let the net income = A = $3,000
and call the Gross amount = G
Tax rate = R = 20% =.20
Social Security = FICA = .062
Medicare = M = .0145
So, Taxes = G X R + G X FICA + G X M = G(R+FICA+M)
Then, Gross amount G = A + taxes = A+G(R+FICA+M)
or,
G - G(R+FICA+M) = A
or,
G(1-R-FICA-M) = A
or,G = A/(1-R-FICA-M)
So,
Gross = 3000/(1-.20-.042-.0145) = 4,146.51
Proof
Taxes = 4146.51 x.20 = 829.30
FICA = 4146.51 x .062 = 257.08
Medicare - 4146.51 x .0145 = 6.12
Summing these all up results in 1,146.51
3,000 plus 1,146.51 = 4146.51
QED
Taxes are paid over depending on what a person owes over a year, to simplify I am using 12 months.
For the OP:
Taxable Income = I
Award = (3000 x 12) = 36000
Taxes owed = C
Taxes paid =D
Without the award the taxpayer would owe:
I/C=D
With the award assuming taxpayer pays no additional taxes but somebody does:
(I + 360000)/C = D
If annual income, awarded income, and taxes owed are the same in both equations you can use the value of C to determine what actual tax bracket a person is in.
I think.
*sigh…taxes paid are the same
If you’re in the “R%” bracket, does that mean your taxes amount to R% of your total income (ignoring, to keep it simple, exemptions and deductions and every other weirdness)?
Or does it mean that you pay a fixed amount on income up to but not including the last marginal increment of your income, and then “R%” only on the last marginal increment of income? (If you earn a whole lot and can’t use the tax tables, isn’t this the way it’s done?)
ETA: And in either case, is there any difference if you are in the lowest bracket? Is the 20% bracket the lowest bracket?
The latter. If you are in the 45% bracket or whatever, it is only the portion of your income that is in that bracket that gets taxed at 45%. If it didn’t work that way people would get less net income when they moved to a higher tax bracket, but that doesn’t happen.
IF he is reasonably likely to also a cost - either because he excluded from the benefit of these, or they somehow affect his other costs (does he pay more for insurances ? ) , maybe you don’t add them back to his income.
Here are the tax brackets for 2014.
So $3,000 a month would put you in the 15% bracket, just shy of the 25% bracket.So the first $9075 are taxed at 10%, then anything over that at 15% (assuming our $36,000 per year annual salary).
I suspect this may be getting more detailed than necessary.
So, as I suspected: Solving the problem that the OP wants does get messy when you have to figure it all out like this, and the straightforward algebraic solution given by Chronos doesn’t work.
Imagine you have a public bus, where a few more passengers get on at each stop, and your task is to compute the total passenger-miles. I picture the problem as being something like that.
Okay, hang tight. I’m trying to work that out. I got an answer that, upon double-checking, comes close (about $136 short of the required annual income), so I’m going to work it out again and see if even that discrepancy goes away, or if it’s within a reasonable rounding error.
BUT, But, but . . .
Just as I feared, it looks like the required annual income to produce this result bumps you up into the next higher tax bracket, so it has to be done again. That’s what I was worried about.
Here’s a quick overview of my thought process:
(1) OP’s stated income of $3000/month must be worked out as an annual figure. That gives $36000/year. (Everyone still with me on that so far?)
(2) Using the 2014 tax bracket list linked by pulykamell above, that seems to put us into 15% bracket.
(3) Working out the required annual taxable income to produce an after-tax income of $36000 (insert detailed algebra, messier than the work shown by Chronos above) comes to something in the vicinity of $43260 (this is the part I seem to be a little off on).
(4) BUT – If your income is in that vicinity, then you’re in the 25% bracket, not the 15% bracket, so it all has to be done over anyway.
That’s where I’m at with it for the moment. Stay tuned . . .
Okay, got it…
I’m doing an over-simplified computations here, filing as a single taxpayer, ignoring any exemptions, deductions, credits, air resistance, etc. We should probably at least assume the standard deduction of $6200 for a single taxpayer. (Again, using the 2014 information linked above.)
Note also, this is for Federal Tax only ignoring any state or other taxes.
The stated income of $3000 a month comes to $36000 a year, which at first glance goes into the 15% bracket. But the required annual pre-tax income to produce the desired post-tax income is rather more than that, and as it turns out, falls into the 25% bracket.
I am finding that an annual income of $42475 is required.
Tax = 5081.25 + 25%(excess over 36900)
For income of $42475 that works out to $6475, leaving an after-tax income of exactly $36000 or $3000 per month.
It seems like, at the very least, we ought to allow our OP to take advantage of the standard deduction if nothing else. For a single taxpayer, that is $6200 for the year 2014.
This means the OP should be able to earn $6200 less for the year (than would be possible without the deduction) and yet still have an after-tax annual income of $36000, or at least approximately something in that neighborhood. At first glance, this looks like it will still be in the 25% bracket, rather than dropping back down to the 15% bracket. (ETA: Wait, there’s something wrong with this, that I still need to think out, but it’s in the right direction anyway.)
(Note: At first thought, you might think the annual pre-tax income should go UP, not down: After all, the OP could earn $6200 more than previously computed, then deduct $6200, and still pay the same tax. But that leaves OP with $6200 more than the specified $36000 annual post-tax income! Remember, in this problem, it’s the post-tax income that we are holding constant. So, to add the $6200 deduction into the equation and still maintain a post-tax income of $36000, we need less pre-tax income to accomplish that!)
I used Excel Solver and came up with $43,986.75. I invite any criticisms but tried to cover everything.
Assuming single status, no special deductions or credits, including at-paycheck taxes, no state tax, working backwards from the solution:
Step 1: Income tax:
-$6200 standard deduction
-$3950 personal exemption
=$33,836.75 taxable income
Tax = 15% bracket (had to find with trial and error = $907.50 + (.15*(33836.75 - 9075) = $4621.762
Part 2: FICA:
Taxable gross income = $43986.75
FICA = (0.062+0.0145)*43986.75 = $3364.986
Total tax = $4621.762 + $3364.986 = $7986.749
Gross income minus tax = $43986.75 - 7986.749 = $36,000
I only rounded at the very end, so it may vary by a tiny bit.
Okay, good for you thelurkinghorror.
I just worked it out again, WITH the standard deduction but WITHOUT including the personal exemption, and WITHOUT taking FICA into account in any way at all. So if thelurkinghorror took all that into account (and assuming he got the right answer, which I didn’t check) then so much the better.
The complication is this: You need to know the correct tax bracket before you can compute the correct tax and thus the required annual income . . . but you also need to know the required annual income before you can know the correct tax bracket. :smack:
The answer is pretty clearly either in the 15% bracket or the 25% bracket. But you have to work it out both ways to find out which one works.
ETA: How much algebra do you need to do yourself to do this in Excel? Is there a pre-programmed formula built-in where you just plug in some numbers? I did it the pen-and-paper-and-algebra way, with only the help of a plain old on-screen calculator. (Yes, I do my math and algebra with a pen. So sue me.)
We still don’t know what tax bracket the OP is in, because he might have other income as well.
Just addition, subtraction, and multiplication. The Solver program I mentioned. It comes with Excel but isn’t installed by default so you turn it on in the options. I just did a few formulas that calculates the number you want (36,000) based on a given starting amount. The Solver tries many, many values for the starting and chooses the best fit to make it end up at or close to 36,000. The AGI ended up being under the 25% bracket so 15% it is although I could do it with 25% as well.