Tax brackets

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Here's a tax bracket question used to illustrate forming formulas.

Question

Suppose tax brackets are as follows: for taxable income between $0 to $32000, the tax rate is 15%; for taxable income between $32000 to $64000, the tax rate is 22%; for taxable income between $64000 and $100000, the tax rate is 25%, for income above $100000, the tax rate is 29%.

a) If x is the taxable income, and T(x) is the tax, find a formula for tax up to $100000.

b) Plot the graph of taxation for taxable income up to $100000.

C) Write, in terms of T(x), the max taxation on the income between $64000 and $100000.

Answer

a) Because the taxation is separated into brackets, we have to separate the answer into "brackets" as well. First of all, for income below $32000, we pay 15% tax. So if income is x < $32000, then tax T(x) = 0.15x. At x = $32000, tax is $32000\times 0.15=\$4800$. Now, if taxable income is between $32000 and $64000, only the portion of income above $32000 has to pay 22% of tax, (and the portion below $32000 would pay 15%, which amounts to $4800), then the tax $T(x)=4800+(x-32000)\times 0.22$. At x = 64000, tax = 4800 + (tax on portion above $32000). The tax on portion over $32000 is $(\$64000-\$32000)\times 0.22=\$7040$. So, at income x = 64000, tax = 4800 + 7040 = 11840. Now, turning to income > 64000, the tax $T(x)=11840+(x-64000)\times 0.25$.

b) If T(x) is on the vertical axis and x is on the horizontal axis, then we plot the following points (0, 0), (32000, 4800), (64000, 11840) and (100000, t), where t = tax you pay at $100000. Plug $100000 into formula $T(x)=11840+(x-64000)\times 0.25=20840$. That is t = 20840.

c) The max tax between $64000 and $100000 is just T(100000) - T(64000) = 20840 - 11840 = 9000.