Let  = Jane's age now
Let  = Joan's age now
Let  = the number of years ago when Jane 
              was half as old Joan is now.
Let  = the number of years ago when Jane 
              was a third as old as Joan is now. 

Jane, who is now , is as old as Joan was
 years ago, . 

That gives us the equation 

Also,  years ago, Jane was , 
and that was half as old as Joan is now, that 
is, half of .

That gives us the equation 

Four years ago, Jane was  and was as 
old as Joan was m years ago, or , 



Also, m years ago, Jane was , which 
was one-third as old as Joan is now, or . 

That gives us the equation 



Can you simplify and solve that system of equation?
If you can't, post again asking how.

Solution:  

Checking:

Jane, who is now 18, is as old as Joan, 
who is now 24, was 6 years ago. That checks because 
she was indeed 18 6 years ago.  

Also, 6 years ago, Jane, who is now 18 was 12, and 
that was half as old as Joan is now, that is, half 
of 24, or 12.  That checks.

Four years ago, Jane, who is now 18 was 14 and was 
as old as Joan, who is now 24, was 10 years ago, or 
14,  That checks 

Also, 10 years ago, Jane was 8, which was one-third 
as old as Joan is now, or 24.  That checks because 8 
is one-third of 24.

She's 18 now, so, in two years she'll be 20.

Edwin