Least Common Denominator (LCD)

The lowest Common Denominator or Least Common Denominator is the Least Common Multiple of the denominators of a set of fractions.

Common denominator : when the denominators of two or more fractions are the same.
Least Common denominator is the smallest of all common denominators.

Why do we need LCD ?
It simplifies addition, subtraction and comparing fraction.

Common Denominator can be simply evaluated by multiplying the denominators. In this case, 3 * 6 = 18

But that may not always be least common denominator, as in this case LCD = 6 and not 18. LCD is actually LCM of denominators.

Examples :

LCD for fractions 5/12 and 7/15 is 60.
We can write both fractions as 25/60 and
28/60 so that they can be added and 
subtracted easily.

LCD for fractions 1/3 and 4/7 is 21.

Example Problem : Given two fractions, find their sum using least common dominator.

Examples:

Input :  1/6  +  7/15    
Output : 19/30
         Explanation : LCM of 6 and 15 is 30. 
         So, 5/30  +  14/30 = 19/30 
Input :  1/3  +  1/6
Output : 3/6
         Explanation : LCM of 3 and 6 is 6. 
         So, 2/6  +  1/6 = 3/6

Note* These answers can be further simplified by Anomalous cancellation.

C++

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// C++ Program to determine
// LCD of two fractions and 
// Perform addition on fractions
#include <iostream>
using namespace std;
  
// function to calculate gcd
// or hcf of two numbers.
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}
  
// function to calculate
// lcm of two numbers.
int lcm(int a, int b)
{
    return (a * b) / gcd(a, b);
}
  
void printSum(int num1, int den1, 
              int num2, int den2)
{
    // least common multiple 
    // of denominators LCD
    // of 6 and 15 is 30.
    int lcd = lcm(den1, den2);
  
    // Computing the numerators for LCD:
    // Writing 1/6 as 5/30 and 7/15 as
    // 14/30
    num1 *= (lcd / den1); 
    num2 *= (lcd / den2); 
  
    // Our sum is going to be res_num/lcd
    int res_num = num1 + num2;
    cout << res_num << "/" << lcd;
}
  
// Driver Code
int main()
{
    // First fraction is 1/6
    int num1 = 1, den1 = 6; 
  
    // Second fraction is 7/15
    int num2 = 7, den2 = 15; 
  
    printSum(num1, den1, num2, den2);
    return 0;
}

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Java

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// Java Program to determine LCD of two
// fractions and Perform addition on 
// fractions
public class GFG {
      
    // function to calculate gcd or 
    // hcf of two numbers.
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
              
        return gcd(b % a, a);
    }
      
    // function to calculate lcm of
    // two numbers.
    static int lcm(int a, int b)
    {
        return (a * b) / gcd(a, b);
    }
      
    static void printSum(int num1, int den1,
                         int num2, int den2)
    {
          
        // least common multiple of 
        // denominators LCD of 6 and 15 
        // is 30.
        int lcd = lcm(den1, den2);
      
        // Computing the numerators for LCD:
        // Writing 1/6 as 5/30 and 7/15 as
        // 14/30
        num1 *= (lcd / den1); 
        num2 *= (lcd / den2); 
      
        // Our sum is going to be res_num/lcd
        int res_num = num1 + num2;
          
        System.out.print( res_num + "/" + lcd);
    }
          
    // Driver code
    public static void main(String args[]) 
    {
          
        // First fraction is 1/6
        int num1 = 1, den1 = 6
  
        // Second fraction is 7/15
        int num2 = 7, den2 = 15
  
        printSum(num1, den1, num2, den2);
    }
}
  
// This code is contributed by Sam007.

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Python3

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# python Program to determine
# LCD of two fractions and 
# Perform addition on fractions
  
# function to calculate gcd
# or hcf of two numbers.
def gcd(a, b):
      
    if (a == 0):
        return b
    return gcd(b % a, a)
  
  
# function to calculate
# lcm of two numbers.
def lcm(a, b):
      
    return (a * b) / gcd(a, b)
  
  
def printSum(num1, den1, 
                  num2, den2):
                  
    # least common multiple 
    # of denominators LCD
    # of 6 and 15 is 30.
    lcd = lcm(den1, den2);
      
    # Computing the numerators
    # for LCD: Writing 1/6 as
    # 5/30 and 7/15 as 14/30
    num1 *= (lcd / den1)
    num2 *= (lcd / den2) 
  
    # Our sum is going to be 
    # res_num/lcd
    res_num = num1 + num2;
    print( int(res_num) , "/" ,
                       int(lcd))
  
# Driver Code
# First fraction is 1/6
num1 = 1
den1 = 6
  
# Second fraction is 7/15
num2 = 7
den2 = 15
printSum(num1, den1, num2, den2);
  
# This code is contributed
# by Sam007

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C#

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// C# Program to determine LCD of two
// fractions and Perform addition on 
// fractions
using System;
  
class GFG {
  
    // function to calculate gcd or 
    // hcf of two numbers.
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
              
        return gcd(b % a, a);
    }
      
    // function to calculate lcm of
    // two numbers.
    static int lcm(int a, int b)
    {
        return (a * b) / gcd(a, b);
    }
      
    static void printSum(int num1, int den1,
                         int num2, int den2)
    {
          
        // least common multiple of 
        // denominators LCD of 6 and 15 
        // is 30.
        int lcd = lcm(den1, den2);
      
        // Computing the numerators for LCD:
        // Writing 1/6 as 5/30 and 7/15 as
        // 14/30
        num1 *= (lcd / den1); 
        num2 *= (lcd / den2); 
      
        // Our sum is going to be res_num/lcd
        int res_num = num1 + num2;
          
        Console.Write( res_num + "/" + lcd);
    }
          
    // Driver code
    public static void Main ()
    {
          
        // First fraction is 1/6
        int num1 = 1, den1 = 6; 
  
        // Second fraction is 7/15
        int num2 = 7, den2 = 15; 
  
        printSum(num1, den1, num2, den2);
    }
}
  
// This code is contributed by Sam007.

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PHP

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<?php
// PHP Program to determine 
// LCD of two fractions and 
// Perform addition on fractions
  
// function to calculate gcd
// or hcf of two numbers.
function gcd($a,$b)
{
    if ($a == 0)
        return $b;
    return gcd($b % $a, $a);
}
  
// function to calculate
// lcm of two numbers.
function lcm($a,$b)
{
    return ($a * $b) / gcd($a, $b);
}
  
function printSum($num1, $den1
                  $num2, $den2)
{
    // least common multiple
    // of denominators
    // LCD of 6 and 15 is 30.
    $lcd = lcm($den1, $den2);
  
    // Computing the numerators for LCD:
    // Writing 1/6 as 5/30 and 7/15 as
    // 14/30
    $num1 *= ($lcd / $den1); 
    $num2 *= ($lcd / $den2); 
  
    // Our sum is going to be res_num/lcd
    $res_num = $num1 + $num2;
    echo $res_num . "/" . $lcd;
}
    // Driver Code
    // First fraction is 1/6
    $num1 = 1;
    $den1 = 6; 
  
    // Second fraction is 7/15
    $num2 = 7;
    $den2 = 15; 
  
    printSum($num1, $den1, $num2, $den2);
  
// This code is contributed by Sam007.
?>

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Output :

19/30

This article is contributed by Shubham Rana. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Improved By : Sam007