1.2.16 有理数，支持加减乘除操作

这题的关键在于：

1. 最大公约数
2. 最小公倍数
3. 通分

public class Rational {
    private final int numerator;
    private final int denominator;
    
    public Rational(int numerator, int denominator) {
        this.numerator = numerator;
        this.denominator = denominator;
    }
    
    /**
     * 和
     * @param b
     * @return
     */
    public Rational plus(Rational b){
        Rational[] operand = rfcd(this, b); 
        int newNumerator = operand[0].numerator + operand[1].numerator;
        int newDenominator = operand[0].denominator;    
        return reduction(new Rational( newNumerator , newDenominator));
    }
    
    /**
     * 差
     * @param b
     * @return
     */
    public Rational minus(Rational b){
        Rational[] operand = rfcd(this, b); 
        int newNumerator = operand[0].numerator - operand[1].numerator;
        int newDenominator = operand[0].denominator;    
        return reduction(new Rational( newNumerator , newDenominator));
    }
    
    /**
     * 积
     * @param b
     * @return
     */
    public Rational times(Rational b){
        Rational[] operand = rfcd(this, b); 
        int newNumerator = operand[0].numerator * operand[1].numerator;
        int newDenominator = operand[0].denominator * operand[0].denominator;
        return reduction(new Rational( newNumerator , newDenominator));
    }
    
    /**
     * 商
     * @param b
     * @return
     */
    public Rational divides(Rational b){
        Rational[] operand = rfcd(this, b);     
        int newNumerator = operand[0].numerator;
        int newDenominator = operand[1].numerator;
        return reduction(new Rational( newNumerator , newDenominator));
    }
    
    public boolean equals(Rational b){
        if(this == b) return true;
        Rational[] operand = rfcd(this, b);
        
        return operand[0].numerator == operand[1].numerator;
    }
    
    public String toString(){
        return "" + this.numerator + "/" + this.denominator;
    }
    
    /**
     * 最大公约数（greatest common divisor）
     * @param p
     * @param q
     * @return
     */
    private int gcd(int p, int q){
        if (q == 0) return p;
        return gcd(q, p%q);
    }
    
    /**
     * 最小公倍数（lowest common multiple）
     * @param a
     * @param b
     * @return
     */
    private int lcm(int a, int b){
        return (a * b) / gcd(a, b);
    }
    
    /**
     * 通分（reduction of fractions to a common denominator）
     * @param a
     * @param b
     * @return
     */
    private Rational[] rfcd(Rational a, Rational b){
        int lcm = lcm(a.denominator, b.denominator);
        int aNewNumerator = a.numerator * (lcm/a.denominator);
        int bNewNumerator = b.numerator * (lcm/b.denominator);      
        return new Rational[]{new Rational(aNewNumerator, lcm), new Rational(bNewNumerator, lcm)};
    }
    
    /**
     * 化简（reduction）
     * @param a
     * @return
     */
    private Rational reduction(Rational a){
        int gcd = gcd(a.numerator, a.denominator);      
        return new Rational(a.numerator/gcd, a.denominator/gcd);
    }
}