Progressive Tax Calculation
Dr. Jones lives in a country with a progressive tax system. That is, he does not pay any taxes on the first $ \$20{,}000$ in income he makes, he pays $5\%$ taxes on the next $\$ 25{,}000 $, he pays $ 10\% $ taxes on the next $ \$35{,}000$ , he pays $15\%$ on the next $\$ 50{,}000 $, and he pays $ 20\% $ on every additional dollar onwards. If Dr. Jones pays $ \$10{,}000$ in taxes, then how much income does he make?
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Solution
If Dr. Jones has an income of $ x $, then the amount of tax is essentially a piecewise function in $ x $. In particular, if we let $ t(x) $ denote the amount of taxes, then $ t(x) = 0 $ when $ 0 \le x \le 20000 $. For $ 20000 \le x \le 45000 $, he pays $$t(x) = 0.05 (x-20000).$$
For $ 45000 \le x \le 80000 $, he pays \begin{align*}
t(x)& = 0.05(45000-20000) + 0.1(x - 45000)\\
& = 1250 + x/10 - 4500.\end{align*}
For $ 80000 \le x \le 130000 $ he pays \begin{align*}
t(x) &= 1250 + 0.1(80000-45000) + 0.15(x - 80000)\\
& = 4750 + 0.15x - 12000.\end{align*}
Finally, if $ x \ge 130000 $, he pays \begin{align*}t(x) &= 4750 + 0.15(130000-80000) + 0.2(x - 130000)\\
& = 12250 + 0.2(x - 130000).\end{align*}We can immediately eliminate the last possibility, since then he would automatically pay at least $ \$12,250$ in taxes. If $x \le 80000$ , then $t(x) \le 1250 + 80000/10 - 4500 = 4750$ . Hence, $80000 \le x \le 130000$. Then, $$10000 = 4750 + 0.15x - 12000 \Longrightarrow x = \boxed{\$115,000}.$$
Freeflows