A hardworking farmer sold his crops at the market. He was set to leave when he met the taxman for the federal government who took half of his gold pieces plus 2 more. A little sad, he next met the state taxman, who took half
of his remaining gold pieces plus two more. Now he was getting depressed. It wasn’t over yet. He next met the county taxman, who took half of his
remaining gold pieces plus two more. Totally discouraged now he spent his two remaining gold pieces and decided to change occupations. How many pieces of gold did he sell his crops for?

Question

A hardworking farmer sold his crops at the market. He was set to leave when he met the taxman for the federal government who took half of his gold pieces plus 2 more.  A little sad, he next met the state taxman, who took half 
of his remaining gold pieces plus two more.  Now he was getting depressed.  It wasn’t over yet.  He next met the county taxman, who took half of his 
remaining gold pieces plus two more.  Totally discouraged now he spent his two remaining gold pieces and decided to change occupations.  How many pieces of gold did he sell his crops for?

mathmale · Answer

Representation is everything in this problem.  I'd begin by letting x represent the total amount of money, in gold pieces, the farmer sold his crop for.  Measure x in gold pieces, e. g., 1 gold piece, 2 gold pieces, and so on.

The dratted taxman takes half of the farmer's gold pieces, plus 2 more:

$$\frac{ x }{ 2 }+2$$ pieces, leaving 
$$x-(\frac{ x }{ 2 }+2)$$
...pieces.   This boils down to $$\frac{ x }{ 2 }-2$$...pieces left.  But then the state taxman takes half of that, plus 2 more.   

And so on. 

The hardest thing here is representing all the unknowns algebraically.

What would you do next?

mathmale · Answer

In the end the poor farmer has just 2 gold pieces left over.  He spends these, so he then has nothing left.   How many gold pieces did he start with?

ohmath123 · Answer

he started with 44 pieces of gold