problem: 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?
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, $$10000 = 4750 + 0.15x - 12000 \Longrightarrow x = \boxed{\$115,000}.$$