头文件：

#ifndef Rational_H
#define Rational_H
class Rational
{
private:
    int numerator;
    int denominator;
public:
    Rational();
    Rational(int , int );
    Rational simplification(Rational&);
    Rational addition(Rational& , Rational& );
    Rational subtraction(Rational& , Rational& );
    Rational multiplication(Rational& a, Rational& b);
    Rational division(Rational& a, Rational& b);
    void showvalue1();//分数形式展现
    void showvalue2();//有理数形式展现
};
#endif

定义类的文件：

#include<math.h>
Rational::Rational() :numerator(1), denominator(1)
{
}
Rational::Rational(int a, int b):numerator(a), denominator(b)
{
    Rational::simplification(*this);
}
Rational Rational::simplification(Rational&a)
{
    int mini = std::min(numerator, denominator);
    for (int i = mini; i > 0; i--)
    {
        if (numerator % i == 0 && denominator % i == 0)
        {
            numerator /= i;
            denominator /= i;
            break;
        }
    }
    return a;
}
Rational Rational::addition(Rational&a, Rational&b)
{
    this->numerator = a.numerator * b.denominator + a.denominator * b.numerator;
    this->denominator = a.denominator * b.denominator;
    Rational::simplification(*this);
    return *this;
}
Rational Rational::subtraction(Rational& a, Rational& b)
{
    this->numerator = a.numerator * b.denominator - a.denominator * b.numerator;
    this->denominator = a.denominator * b.denominator;
    Rational::simplification(*this);
    return *this;
}
Rational Rational::multiplication(Rational& a, Rational& b)
{
    this->numerator = a.numerator * b.numerator;
    this->denominator = a.denominator * b.denominator;
    Rational::simplification(*this);
    return *this;
}
Rational Rational::division(Rational& a, Rational& b)
{
    this->numerator = a.numerator * b.denominator;
    this->denominator = a.denominator * b.numerator;
    Rational::simplification(*this);
    return *this;
}
void Rational::showvalue1()
{
    std::cout << numerator << "/" << denominator<<std::endl;
}
void Rational::showvalue2()
{
    double value = (double)numerator / denominator;
    std::cout << value << std::endl;
}

一个demo：

#include"head.h"
int main()
{
    //测试
    Rational x(16, 12);
    Rational y(4, 9);
    Rational a1;
    Rational a2;
    Rational a3;
    Rational a4;
    a1.addition(x, y);
    a2.subtraction(x, y);
    a3.multiplication(x, y);
    a4.division(x, y);
    a1.showvalue1();
    a2.showvalue1();
    a3.showvalue1();
    a4.showvalue1();
    a1.showvalue2();
    a2.showvalue2();
    a3.showvalue2();
    a4.showvalue2();
}