pretax income algebra question

Factual Questions
1

Although I haven’t taken math in over ten years, I’m embarrassed at not being able to figure this one out.

I determined that my minimum aftertax income required to meet expenses is roughly $32000. In order to determine pretax salary requirements it seems like the equation could be expressed like this:

aftertax income=gross income - tax
tax= taxrate(gross income-(standard deduction+personal exemption))
So (given standard deduction +personal exemption for single being 7800 and the tax rate approximately 25%):

32000=x-.25(x-7800)
32000=x-.25x-1950
32000=.75x-1950
33950=.75x
x=45266.67

However, when subtract the standard deduction from the answer (result = $37466) and then go to the 2003 tax table and subtract out the appropriate tax ($6176) from the gross I get $39090 instead of 32000.

Is the algebra wrong or am I figuring the tax wrong? If I guesstimate and figure that I’m going to have to pay about $5000 in taxes the $37000 seems a lot closer than $45000 for a pretax income.

2

In the variable equations you’ve given, you have the deduction stuff inside the parentheses, which means it is being multiplied by .25 giving you, apparently, 1950. But when you are back calculating, you are using the 7800 number. I’m not up on taxcode or nothing, but I’d think you’d want the deductions outside the brackets in the original equations.

3

Go to this link and look at puzzle number 5.

http://www.1728.com/puzzle.htm

4

I think I get puzzle # 5, but it doesn’t deal with deductions. So where the puzzle puts in the .325 tax rate, I don’t know what it would be in mine (I don’t think it’s .25). Could you tell me what tax rate to plug into the equation?

Also, just because I’m frustrated with my apparently less than 9th grade math level, is there a math whiz who can tell me where I went wrong with my algebra?

5

Here’s your mistake. 32000 = x - .25x + 1950
That gives you:
32000 = .75x + 1950
30050 / .75 = x
x = 40067

6

Your mistake is that there is not neccesarily a single tax rate for a given amount of income. For example, the first 0 to x $s of income are taxed at 10%, the next x+1 to y $s are taxed at 15%, the next y+1 to z are taxed at 20%, and so on. You cant’t just multiply your income by a single rate to determine your tax.

The highest tax rate you pay is called your marginal tax rate.