#include <cassert>
#include <iostream>
#include <numeric>

struct Rational {
  int numerator;
  int denominator;
};

// Garante que:
// - r.denominator > 0
// - std::gcd(r.numerator, r.denominator) == 1
void normalize(Rational &r) {
  // Se denominador fornecido e zero, nao ha o que fazer.
  assert(r.denominator != 0);

  // Garante que o denominador e positivo.
  if (r.denominator < 0) {
    r.numerator = -r.numerator;
    r.denominator = -r.denominator;
  }

  // Elimina fatores comuns no numerador e denominador.
  auto common = std::gcd(r.numerator, r.denominator);
  r.numerator /= common;
  r.denominator /= common;
}

// Calcula a * b
Rational multiply(Rational a, Rational b) {
  Rational result;
  result.numerator = a.numerator * b.numerator;
  result.denominator = a.denominator * b.denominator;
  normalize(result);
  return result;
}

// Calcula a / b
Rational divide(Rational a, Rational b) {
  Rational result;
  result.numerator = a.numerator * b.denominator;
  result.denominator = a.denominator * b.numerator;
  normalize(result);
  return result;
}

// Calcula a + b
Rational add(Rational a, Rational b) {
  Rational result;
  result.numerator = a.numerator * b.denominator + a.denominator * b.numerator;
  result.denominator = a.denominator * b.denominator;
  normalize(result);
  return result;
}

// Calcula a - b
Rational subtract(Rational a, Rational b) {
  Rational result;
  result.numerator = a.numerator * b.denominator - a.denominator * b.numerator;
  result.denominator = a.denominator * b.denominator;
  normalize(result);
  return result;
}

// Calcula -a
Rational negative(Rational a) {
  Rational result;
  result.numerator = -a.numerator;
  result.denominator = a.denominator;
  return result;
}

// Verifica se a == b
bool are_equal(Rational a, Rational b) {
  return a.numerator == b.numerator && a.denominator == b.denominator;
}

// Ajusta numerador e denominador de r.
void set(Rational &r, int numerator, int denominator) {
  r.numerator = numerator;
  r.denominator = denominator;
  normalize(r);
}

void show(std::ostream &os, Rational r, std::string sep = "\n") {
  os << r.numerator << "/" << r.denominator << sep;
}

int main(int, char *[]) {
  Rational a, b, c;
  set(a, 1, 4);
  set(b, 3, 7);

  c = multiply(a, b);
  show(std::cout, c);

  c = divide(a, b);
  show(std::cout, c);

  c = add(a, b);
  show(std::cout, c);

  c = subtract(a, b);
  show(std::cout, c);

  set(c, 0, 1);
  for (int i = 0; i < 10; ++i) {
    c = add(c, multiply(a, b));
    show(std::cout, c, ", ");
  }
  std::cout << std::endl;
}