Maximum rational number (or fraction) from an array

Given rational numbers, the task is to find the maximum rational number.

Examples:

Input : ra_num = {{1, 2},
              {2, 3},
              {3, 4},
              {4, 5}};
Output : 4 5

Input : ra_num = {{10, 12},
              {12, 33},
              {33, 14},
              {14, 15}};
Output : 33 14

A simple solution is to find float values and compare the float values. The float computations may cause precision errors. We can avoid them using the below approach.
Say numbers are 1/2, 2/3, 3/4, 4/5
First take an LCM of (2, 3, 4, 5) which is the denominator of all rational numbers. So the LCM of this is 60, then divide with all denominator’s and multiple with all numerators, so the value of numerators are (30, 40, 45, 48) 
Then find the max between these rational numbers. So here the last numerator is max then print the last rational number, which is 4/5.

C++

// CPP program to find the maximum rational
// number in an array.
#include <bits/stdc++.h>
using namespace std;

struct Rational {

    // numerator and Denominator
    int nume, deno;
};

// here we find the Denominator LCM
int lcmOfDenominator(vector<Rational> ra_num)
{
    // get the first Denominator as lcm
    int lcm = ra_num[0].deno;
    int i;

    // find the lcm of all relational
    // number Denominator
    for (i = 1; i < ra_num.size(); i++)
        lcm = (lcm * (ra_num[i].deno)) /
                 __gcd(lcm, ra_num[i].deno);


    // return the lcm
    return lcm;
}

int maxRational(vector<Rational> ra_num)
{
    // take a temp array for find
    // maximum numerator after multiple
    int temp[ra_num.size()] = { 0 };

    // get here the lcm of all rational
    //number denominator
    int lcm = lcmOfDenominator(ra_num);

    // take maximum for get maximum index
    int maximum = 0;
    int maximumind = 0;

    // find the index which contain maximum value
    for (int i = 0; i < ra_num.size(); i++) {

        // divide lcm with denominator
        // and multiple with numerator
        temp[i] = (ra_num[i].nume) *
                  (lcm / ra_num[i].deno);

        // get the maximum numerator
        if (maximum < temp[i]) {
            maximum = temp[i];
            maximumind = i;
        }
    }

    // return index which contain
    // maximum rational number
    return maximumind;
}

int main()
{
    // given rational number
    vector<Rational> ra_num = { { 1, 2 },
                                { 2, 3 },
                                { 3, 4 },
                                { 4, 5 } };

    // get the index which contain maximum value
    int index_max = maxRational(ra_num);

    // print numerator and denominator
    cout << ra_num[index_max].nume << " "
         << ra_num[index_max].deno << "\n";
}

Java

// Java program to find the maximum rational
// number in an array.
import java.util.*;

class GFG
{

static class Rational
{

    // numerator and Denominator
    int nume, deno;

    public Rational(int nume, int deno)
    {
        this.nume = nume;
        this.deno = deno;
    }

};

// here we find the Denominator LCM
static int lcmOfDenominator(Vector<Rational> ra_num)
{
    // get the first Denominator as lcm
    int lcm = ra_num.get(0).deno;
    int i;

    // find the lcm of all relational
    // number Denominator
    for (i = 1; i < ra_num.size(); i++)
        lcm = (lcm * (ra_num.get(i).deno)) /
                __gcd(lcm, ra_num.get(i).deno);

    // return the lcm
    return lcm;
}

static int maxRational(Vector<Rational> ra_num)
{
    // take a temp array for find
    // maximum numerator after multiple
    int []temp = new int[ra_num.size()];

    // get here the lcm of all rational
    //number denominator
    int lcm = lcmOfDenominator(ra_num);

    // take maximum for get maximum index
    int maximum = 0;
    int maximumind = 0;

    // find the index which contain maximum value
    for (int i = 0; i < ra_num.size(); i++)
    {

        // divide lcm with denominator
        // and multiple with numerator
        temp[i] = (ra_num.get(i).nume) *
                  (lcm / ra_num.get(i).deno);

        // get the maximum numerator
        if (maximum < temp[i])
        {
            maximum = temp[i];
            maximumind = i;
        }
    }

    // return index which contain
    // maximum rational number
    return maximumind;
}

static int __gcd(int a, int b)
{
    if (b == 0)
        return a;
    return __gcd(b, a % b);
}

// Driver Code
public static void main(String[] args)
{
    // given rational number
    Vector<Rational> ra_num = new Vector<Rational>();
        ra_num.add(new Rational( 1, 2 ));
    ra_num.add(new Rational( 2, 3 ));
    ra_num.add(new Rational( 3, 4 ));
    ra_num.add(new Rational( 4, 5 ));

    // get the index which contain maximum value
    int index_max = maxRational(ra_num);

    // print numerator and denominator
    System.out.println(ra_num.get(index_max).nume +
                 " " + ra_num.get(index_max).deno);
    }
}

// This code is contributed by Princi Singh

Python3

# Python3 program to find the maximum rational
# number in an array.
class Rational:

    def __init__(self, nume, deno):

        # Numerator and Denominator
        self.nume = nume
        self.deno = deno

def computeGCD(x, y):

    while(y):
        x, y = y, x % y

    return x

# Here we find the Denominator LCM
def lcmOfDenominator(ra_num):

    # Get the first Denominator as lcm
    lcm = ra_num[0].deno

    # Find the lcm of all relational
    # number Denominator
    for i in range(1, len(ra_num)):
        lcm = ((lcm * (ra_num[i].deno)) //
       computeGCD(lcm, ra_num[i].deno))

    # return the lcm
    return lcm

def maxRational(ra_num):

    # Take a temp array for find
    # maximum numerator after multiple
    temp = [0 for i in range(len(ra_num))]

    # Get here the lcm of all rational
    # number denominator
    lcm = lcmOfDenominator(ra_num)

    # Take maximum for get maximum index
    maximum = 0
    maximumind = 0

    # Find the index which contain
    # maximum value
    for i in range(len(ra_num)):

        # Divide lcm with denominator
        # and multiple with numerator
        temp[i] = ((ra_num[i].nume) *
            (lcm // ra_num[i].deno))

        # Get the maximum numerator
        if (maximum < temp[i]):
            maximum = temp[i]
            maximumind = i

    # Return index which contain
    # maximum rational number
    return maximumind

# Driver code
if __name__=="__main__":

    # Given rational number
    ra_num = []
    ra_num.append(Rational(1, 2))
    ra_num.append(Rational(2, 3))
    ra_num.append(Rational(3, 4))
    ra_num.append(Rational(4, 5))

    # Get the index which contain maximum value
    index_max = maxRational(ra_num)

    # Print numerator and denominator
    print(str(ra_num[index_max].nume) + " " +
          str(ra_num[index_max].deno))

# This code is contributed by rutvik_56

C#

// C# program to find the maximum rational
// number in an array.
using System;
using System.Collections.Generic;

class GFG
{

public class Rational
{

    // numerator and Denominator
    public int nume, deno;

    public Rational(int nume, int deno)
    {
        this.nume = nume;
        this.deno = deno;
    }

};

// here we find the Denominator LCM
static int lcmOfDenominator(List<Rational> ra_num)
{
    // get the first Denominator as lcm
    int lcm = ra_num[0].deno;
    int i;

    // find the lcm of all relational
    // number Denominator
    for (i = 1; i < ra_num.Count; i++)
        lcm = (lcm * (ra_num[i].deno)) /
                __gcd(lcm, ra_num[i].deno);

    // return the lcm
    return lcm;
}

static int maxRational(List<Rational> ra_num)
{
    // take a temp array for find
    // maximum numerator after multiple
    int []temp = new int[ra_num.Count];

    // get here the lcm of all rational
    //number denominator
    int lcm = lcmOfDenominator(ra_num);

    // take maximum for get maximum index
    int maximum = 0;
    int maximumind = 0;

    // find the index which contain maximum value
    for (int i = 0; i < ra_num.Count; i++)
    {

        // divide lcm with denominator
        // and multiple with numerator
        temp[i] = (ra_num[i].nume) *
                  (lcm / ra_num[i].deno);

        // get the maximum numerator
        if (maximum < temp[i])
        {
            maximum = temp[i];
            maximumind = i;
        }
    }

    // return index which contain
    // maximum rational number
    return maximumind;
}

static int __gcd(int a, int b)
{
    if (b == 0)
        return a;
    return __gcd(b, a % b);
}

// Driver Code
public static void Main(String[] args)
{
    // given rational number
    List<Rational> ra_num = new List<Rational>();
                   ra_num.Add(new Rational( 1, 2 ));
                   ra_num.Add(new Rational( 2, 3 ));
                   ra_num.Add(new Rational( 3, 4 ));
                   ra_num.Add(new Rational( 4, 5 ));

    // get the index which contain maximum value
    int index_max = maxRational(ra_num);

    // print numerator and denominator
    Console.WriteLine(ra_num[index_max].nume +
                " " + ra_num[index_max].deno);
    }
}

// This code is contributed by PrinciRaj1992

Javascript

<script>
      // JavaScript program to find the maximum rational
      // number in an array.

      class Rational {
        constructor(nume, deno) {
          this.nume = nume;
          this.deno = deno;
        }
      }

      // here we find the Denominator LCM
      function lcmOfDenominator(ra_num) {
        // get the first Denominator as lcm
        var lcm = ra_num[0].deno;
        var i;

        // find the lcm of all relational
        // number Denominator
        for (i = 1; i < ra_num.length; i++)
          lcm = (lcm * ra_num[i].deno) / __gcd(lcm, ra_num[i].deno);

        // return the lcm
        return lcm;
      }

      function maxRational(ra_num) {
        // take a temp array for find
        // maximum numerator after multiple
        var temp = new Array(ra_num.Count);

        // get here the lcm of all rational
        //number denominator
        var lcm = lcmOfDenominator(ra_num);

        // take maximum for get maximum index
        var maximum = 0;
        var maximumind = 0;

        // find the index which contain maximum value
        for (var i = 0; i < ra_num.length; i++) {
          // divide lcm with denominator
          // and multiple with numerator
          temp[i] = ra_num[i].nume * (lcm / ra_num[i].deno);

          // get the maximum numerator
          if (maximum < temp[i]) {
            maximum = temp[i];
            maximumind = i;
          }
        }

        // return index which contain
        // maximum rational number
        return maximumind;
      }

      function __gcd(a, b) {
        if (b === 0) return a;
        return __gcd(b, a % b);
      }

      // Driver Code
      // given rational number
      var ra_num = [];
      ra_num.push(new Rational(1, 2));
      ra_num.push(new Rational(2, 3));
      ra_num.push(new Rational(3, 4));
      ra_num.push(new Rational(4, 5));

      // get the index which contain maximum value
      var index_max = maxRational(ra_num);

      // print numerator and denominator
      document.write(ra_num[index_max].nume + " " + ra_num[index_max].deno);
    </script>

Output:

4 5

Time Complexity: O(N*log(K)) where N is the size of the given array and K can be the largest denominator of the array.

Auxiliary Space: O(N)