[Math] Calculate gross salary when knowing the net salary and tax brackets

arithmeticfinance

Assume the annually gross salary is $100,000.

Tax brackets:

  1. 0 – 50K – 10% = 5K in taxes
  2. 50K – 70K – 20% = 4K in taxes
  3. 70K – $90K – 30% = 6K in taxes
  4. 90K and up – 40% = 4K in taxes

The income tax on 100K would be 19K.

So the net salary would be 81K.

However, knowing only the net salary (81K) and the tax brackets (1-4) how can I find out what was the gross salary?

Thank you very much.

UPDATE: Would the same rules apply if I have 2 level of brackets? For instance Federal and State income tax. Lets assume that the brackets are the same and Federal and State charge you the same amount on yeah tax bracket.

UPDATE 2: Actually I will combine Federal and State tax into one set of brackets and apply the same rule as it was one set of brackets.

Best Answer

First, determine the maximum net for each bracket:

  1. If your gross is 0-50,000, then your net is 0-45,000.

  2. If your gross is 50,001-70,000, then your net is 45,001-61,000 (the most you will pay in taxes is 5,000 for the first 50,000, and 4,000 for the next 20,000).

  3. If your gross is 70,001-90,000, then your net is 61,001-75,000 (the most you will pay in taxes is 5,000 for the first 50,000, 4,000 for the next 20,000, and 6,000 for the final 20,000, for a total of 15,000 in taxes).

  4. If your gross is 90,001 and up, your net is 75,001 and up.

So, if the net salary is 81,000, then you are in the final bracket. You've earned 75,000 for the first 90,000; and you've earned a further 6,000 more net.

If $x$ is the amount over 90,000 that you earned, then you know that $0.6x=6,000$ (since you keep 60% of the amount over 90,000). Solving for $x$ gives $x=\frac{6000}{0.6} = 10,000$, so the gross salary was $\$$100,000.

Indeed, if your gross is 100,000, then you pay 5,000 of the first 50,000 and keep 45,000; you pay 4,000 of the next 20,000, and keep 16,000 (you've paid 9,000 in taxes and kept 61,000 in net); you pay 6,000 of the next 20,000 in taxes, and keep 14,000 (your running total is 15,000 in taxes and 75,000 in net); and you pay 4,000 in taxes for the next 10,000 and keep 6,000 (your final total: 19,000 in taxes, 81,000 in net).

Once you know what tax bracket the final total lies in, you can figure out how much "after the last dividing line" he originally earned.

For example, say the net salary is 67,500. This corresponds to a gross salary in the 70,000-90-000 bracket; it is exactly 6,500 over the final net if you made 70,000, so if $x$ is the amount over 70,000, as you pay 30% of $x$ in taxes you must have $6,500=0.7x$. Solving for $x$ gives $x=9285.71$, so the gross would be $70,000+x = 79,285.71$.

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