Find pattern in two numbers

Question

If I'm given two sets of numbers, how can I find out what the fixed amount fee is and what's the variable? Example

\$100 - taxes and fees (\$19.7 or 19.7%) = \$119.7

\$200 - taxes and fees (\$33.7 or 16.85%) = \$233.7

The formula is

(Price + Fee)*Tax

I'm trying to solve for Fee and tax

Answer 1

Let $f$ be the fee and $t$ be the tax. I believe your formula should be the final price is (Price+Fee)(1+tax). Just multiplying by the tax rate gives you the tax, but not the final cost. Plug in your data and you have two equations in two unknowns $$(100+f)(1+t)=119.7\\(200+f)(1+t)=233.7\\ f+100t+ft=19.7\\f+200t+ft=33.7\\ 100t=14\\t=0.14\\100+f=\frac {119.7}{1.14}=105\\ f=5$$ where the fifth line comes from subtracting the two above.

Answer 2

You can solve it by using a system of two equations with two unknowns fee $f$ and tax $t-1$: $$ \begin{cases} (100 + f) t = 119.7 \\ (200 + f) t = 233.7 \end{cases} $$

Can you take it from here or do you need more help?