Let X = Marty's age now

Let Y = Mother's age now

Then 

Marty's age five years age = (X-5)

Mother's age five years ago = (Y-5)

1/3 of Mother's age now = 

1/5 of Mother's age five years ago = 

Marty's age = 1/3 of Mother's age now
  
     X      =              

Marty's age five years ago = 1/5 of Mother's age five years ago 

         (X-5)             =     

     
So we have these two equations 



Multiply both sides of the first equation by 3 to clear of fractions.
Multiply both sides of the first equation by 5 to clear of fractions.



Canceling clears fractions:





Simplifying further:



Simplifying further:



Since the first equations tells us that Y = 3X we can replace the Y
in the second equation 



by 3X







So Marty is 10 years old.  However we need to find her
Mothers age so we can check.  We use

Y = 3X

Y = 3(10)

Y = 30

So her mother is 30.

Let's check:

Marty is 1/3 as old as her mother.

Indeed 10 is 1/3 of 30

Five years ago she was only 1/5 of the age of her mother

Five years ago se was 5 and he mother was 25, and indeed
5 is 1/5 of 25.

So we know we are right.

Edwin