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

// VERSÃO 3!
class Rational {
  int _numerator;
  int _denominator;

  // Garante que:
  // - r.denominator > 0
  // - std::gcd(r.numerator, r.denominator) == 1
  void _normalize();

public:
  // Cria um racional com numerador e denominador especificados.
  Rational(int numerator = 0, int denominator = 1)
      : _numerator(numerator), _denominator(denominator) {
    _normalize();
  }

  // Calcula a * b.
  friend Rational multiply(Rational a, Rational b);
  // Calcula a / b.
  friend Rational divide(Rational a, Rational b);
  // Calcula a + b.
  friend Rational add(Rational a, Rational b);
  // Calcula a - b.
  friend Rational subtract(Rational a, Rational b);
  // Calcula -a.
  friend Rational negative(Rational a);

  // Verifica se a == b.
  friend bool are_equal(Rational a, Rational b) {
    return a._numerator == b._numerator && a._denominator == b._denominator;
  }

  // Acesso (apenas leitura) a numerador e denominador.
  int numerator() const { return _numerator; }
  int denominator() const { return _denominator; }
};

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

  // Garante que o denominador e positivo.
  if (_denominator < 0) {
    _numerator = -_numerator;
    _denominator = -_denominator;
  }

  // Elimina fatores comuns no numerador e denominador.
  auto common = std::gcd(_numerator, _denominator);
  _numerator /= common;
  _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;
  result._normalize();
  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;
  result._normalize();
  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;
  result._normalize();
  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;
  result._normalize();
  return result;
}

// Calcula -r.
Rational negative(Rational r) {
  Rational result;
  result._numerator = -r._numerator;
  result._denominator = r._denominator;
  return result;
}

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

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

  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);

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