For a fraction the number of counters is called the numerator and the number of boards is called the denominator.
Numerator comes from to numerate or to enumerate meaning to count.
Denominator comes from to name, such as religious groups have different denominations such as Methodist, Baptist or Quaker, ie have different names. Coins have different denominations such as 5p or 10p or 50p. We saw in New Puzzle, New Numbers that the name for a fraction, its denominator, comes from the number of boards.
In the above case there are three boards and the name of the fractions being counted is Thirds.
We numerate or count two Thirds.
So Two is the numerator because is counts how many Thirds.
Three is the denominator because it gives the name to what is being counted, that is Thirds.
For Five is the numerator and Seven is the denominator.