(* Fifth version of Rational that reimplements the type using an int * int.
   This change would be invisible (opaque) to a client of the structure. *)

module type RATIONAL =
sig
   type rational
   exception Not_a_rational
   val make_fraction : int * int -> rational
   val whole : int -> rational
   val add : rational * rational -> rational
   val to_string : rational -> string
end

module Rational : RATIONAL =
struct
    type rational = int * int
    exception Not_a_rational

    let rec gcd(x, y) =
        if x < 0 || y < 0 then gcd(abs(x), abs(y))
        else if y = 0 then x
        else gcd(y, x mod y)

    let rec reduce_rational(a, b) =
        if b < 0 then reduce_rational(-a, -b)
        else let d = gcd(a, b)
             in (a/d, b/d)

    let make_fraction(a, b) = 
        if b = 0 then raise Not_a_rational
        else reduce_rational(a, b)

    let whole(a) = (a, 1)

    let add((a, b), (c, d)) = reduce_rational(a * d + c * b, b * d)

   let to_string(a, b) =
        if b = 1 then string_of_int(a)
        else string_of_int(a) ^ "/" ^ string_of_int(b)
end