signature Rational =
sig
  type R
  exception DenomIsZero
  val makerat: int*int -> R
  val numer: R -> int
  val denom: R -> int
  val add: R*R -> R
end;

structure Rat : Rational  =
struct
  exception DenomIsZero;
  datatype R = ratify of int*int;
  fun gcd(a,b) = if (b=0) then a else gcd(b,a mod b);
  fun makerat(p,q) =
      if (q=0) then raise DenomIsZero
      else 
         let
           val g = gcd(p,q);
         in
           ratify(p div g,q div g)
         end;
  fun numer(ratify(p,_)) = p;
  fun denom(ratify(_,q)) = q;
  fun add(a:R,b:R) =
         makerat(numer(a)*denom(b)+numer(b)*denom(a),denom(a)*denom(b));
end;