Calculate Tithe from Income

Question

Hi I need help calculating the amount I need to donate as tithe (1/10 of income) every year.

Say my annual income before tax is $50000 and the tax rate is 30%. Which means my income after tax is $50000 x 0.7 = $35000. So I will donate $3500 as tithe.

Now since donation is tax free in my country, any donation >= $2 can claim the tax back. Which means I can claim back 30% of $3500, which is $1050.

That $1050 now becomes my income and I should again donate 1/10 of it as tithe, which is $105.

The same process repeats again after claiming tax back for that $105 donation until the donation is < $2 where it is too little to be claimable.

Without doing the whole process repeatedly over multiple tax claims and donations, I want to know the exact amount I should donate as tithe in one go (obviously will be > $3500), so that the tax claimed back is kept for myself.

Please answer with formula of variables rather than exact amount used in this example.

Answer 1

We care about three quantities: the amount of money you make before taxes $(G)$, the amount you offer as tithe $(T$), and the amount you pay in taxes $(X)$.

You don't pay any taxes on the amount you give away as tithe, but you do pay taxes on the rest. Therefore, with a tax rate of 30%, we have in general: $$X = 0.3\cdot(G-T)$$

You want to give away 10% of your after-tax income as tithe: $$T = 0.10 (G-X) $$

Let's combine these two equations to eliminate $X$ so that we have:

$$T = 0.10G - 0.10(0.3G-0.3T)$$ $$T = 0.07G + 0.03T$$ $$0.97T = 0.07G$$ $$\boxed{T = \frac{7}{97}G}$$ $$T \approx 0.0721 \cdot G$$

For example, if you make \$50000 before taxes, you would pay around \$3608.24 in tithe—slightly more than the \$3500 you would pay if you didn't use the recursive formula.

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Note: If I understand your situation correctly, we can ignore the \$2 limit as long as you're giving away the entire tithe $T$ all in one payment and its total value is greater than \$2. For example, if you just give your entire tithe of \$3608.24 all at once, it's not taxable.

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And in the general problem where tithes are untaxed and you want to recursively pay $p$ percent of your income as tithe given $q$ percent tax rate, the same algebraic reasoning shows you should pay

$$T = \frac{p(1-q)}{1-pq}G$$ which agrees with our earlier answer.

Answer 2

You want to find: $$3500+105+3.15+\cdots=\frac{3500}{1-0.3\cdot 0.1}=\frac{3500}{0.97}\approx 3608.247.$$ Note: $T_{n+1}=0.03T_n, T_1=3500$.

Answer 3

Let $G$=gross income and $T$=tithe.

Your after-tax income, for purposes of computing the tithe is $$.7G+.3T,$$ and the tithe is $10$ percent of this: $$T=.07G+.03T\implies .97T=.07G\implies\\ \boxed{T=\frac{7}{97}G\approx .0722G}$$

This gives $T=\$3608.25\text{ when } G=\$50000.$