How to calculate income needed with recursive taxes?

I'm looking for a formula to calculate how much money I need to make to cover expenses and taxes, which doesn't seem simple because of recursion.

Let's assume the tax rate is 20% on all income (no deductions), and expenses of $10,000/mo.
I'd need to make $10k to cover expenses, so I'd owe $2,000/mo. in taxes on that income.
To cover expenses *and* taxes, I'd need to earn $10+2k = $12k.

But if I'm earning $12,000/mo., then my taxes aren't $2k/mo., they're $12k x 20% = $2400/mo.
Okay, so I need to earn $10,000 + $2400 = $12,400 to cover expenses + taxes.

But if I'm earning $12,400/mo. then my taxes aren't $2400/mo., they're $12,400 x 20% = $2480/mo.
Okay, so I need to earn $10,000 + $2480 = $12,480.

And so on and so on.

Once taxes hit $2500 then the recursion stops, because the extra taxes on the extra income is less than a dollar.

One way to solve this problem is to manually calculate each iteration. But that's clumsy. Surely there's a general formula I can apply?

10,000 / (1 - .20) = 12,500

Thank you, Ace2.

Of course, now I'm realizing it's simple algebra:

Quote: math

expenses + tax = income

expenses + (20% * income) = income

expenses = income - (20% * income)

expenses = 0.8 income

expenses ÷ 0.8 = income

10,000 ÷ 0.8 = 12,500


income - expenses = taxes

$12,500 - $10,000 = $2500

For years I've been using a similar formula, but somehow spaced that I could use the same concept for the above case. If I have the total price, and want to figure what the price was before 8.25% tax:

Quote: math

price + (price * 8.25%) = total

1.0825 price = total

total ÷ 1.08 = price

Quote: MichaelBluejay

Thank you, Ace2.

Of course, now I'm realizing it's simple algebra:

LOL

I had a similar problem when I was bidding a job and the salesperson commission was variable since it was a percent. I wanted to make X in profit bit knew I could not just add Y percent because it made X higher. So I used simple algebra. It amazed my second in command as his math skills were shall we say weak. Then for the maybe the first time I realized why they made us take algebra.

It would have ended there but years later I was working in call center hell and doing car loans. We made small talk as part of the process and an algebra teacher calls. Somehow she says how she cannot make the kids understand why the need to learn it. So I tell the story and get a bunch of "uh huhs" as she is listening. Then she says, "Hold on a sec. TELL ME THAT AGAIN BUT TALK SLOWER SO I CAN WRITE THIS DOWN!"

Hopefully I helped at least one kid get it.

So one out of a hundred or maybe a thousand use algebra in their lifetimes and the rest of us should suffer. Got it. A course on money management, like using a checking account, auto loans and home/ rent finances would be infinitely more useful.

Thing with math is you can usually hire someone to do it for you pretty cheap.
As long as you can write a check, you can get math done for you.

Quote: billryan

Thing with math is you can usually hire someone to do it for you pretty cheap.
As long as you can write a check, you can get math done for you.

True of a lot of things.

Math is how things work. Underlies most scientific and technological progress.

The pour souls that were forced into sitting through algebra they didn’t understand and don’t use, are the necessary sacrifice on the alter of finding the souls that understood algebra and excelled in math, finding professions today where they use it for the betterment of science and technology.

This reminds me of when I did my parents' taxes in the 1990s - their social security and total income meant that they couldn't deduct the full amount of their IRA deposits, but the IRA deduction was based on the AGI that included how much of their social security income was taxable - and the taxable social security amount was based on what their IRA deduction was.

IIRC, what I had to do was:
1. Calculate the adjusted gross income (AGI) under the assumption that 85% of the social security income was taxable (the maximum amount) and there was no IRA;
2. Recalculate the AGI after an IRA deduction using the step 1 AGI;
3. Recalculate the amount of taxable social security income using the step 2 AGI;
4. Calculate the actual AGI after an IRA deduction using the step 3 AGI.

I think it's good to learn some math and science, even if we don't use it, as appreciation for the civilization that we're enjoying. We have cell phones, Netflix, Internet, cars, airplanes, comfortble buildings, etc., all because of other people smarter than most of us who discovered/invented stuff. I don't think we should take that for granted.

On the other hand, high school definitely should also cover things that are more useful / practical, like personal finance, avoiding scams, and psychology (e.g., how not to recognize when you're getting sucked into a cult, cognitive bias, etc. Not everyone takes psychology in college, and the sooner you learn some of that stuff, the better.)

A common financial math mistake is, for instance, if an investment goes down 50%, people think it needs to go up only 50% to fully recover.

Quote: DeMango

So one out of a hundred or maybe a thousand use algebra in their lifetimes and the rest of us should suffer. Got it. A course on money management, like using a checking account, auto loans and home/ rent finances would be infinitely more useful.

I'd still have people take Algebra and Geometry but then split people to a math college track and a more business oriented math. Investment calculation and a helping of basic stats.

The problem I always had and saw with Algebra was how abstract it was. B left an hour after A, how fast does B need to go to catch A? Who cares. tell B to be ready to leave when her husband A says they have to leave and no problem!

If it was made more real I think more people would get it. I am fairly convinced if I had a partner to do some animation and a math guy for the math end I could make a YT-like course people would get the first time. Life is too short to get it done.

Quote: AZDuffman

If it was made more real I think more people would get it. I am fairly convinced if I had a partner to do some animation and a math guy for the math end I could make a YT-like course people would get the first time..

Agreed. I remember my first and only probability & statistics course in college. I didn’t really understand or care until we started calculating the probability of poker hands...then it clicked.