Common Denominator (LCD) Calculator

LCD calculator uses two or more fractions, integers or mixed numbers and calculates the least common denominator, i.e. the smallest positive integer which is divisible by each denominators of these numbers. It is an online mathematical tool specially programmed to find out the least common denominator for fractions with different or unequal denominators.
It is necessary to follow the next steps:

1. Enter two or more numbers in the box. These numbers must be fractions, integers or mixed numbers and may be separated by commas. The numbers can be copied from a text document or a spreadsheet;
2. Press the "GENERATE WORK" button to make the computation;
3. Common denominator calculator will give the LCD of two or more different or unequal denominators of fractions.

Input: Two or more fractions, integers or mixed numbers;
Output: The least common denominator is an integer number.

The common denominator calculator determines the least common denominator as the least common multiple of two or more integers using the prime factorization.

What is Least Common Denominator?

To replace unlike fractions by equivalent like fractions, it is necessary to find a common denominator. A problem of finding the common denominator of two or more fractions is the same as the problem of finding the least common multiple (LCM) of the denominators of these fractions. If the denominators of the given fractions are equal to each other, than the LCD of these fractions is the denominator of these fractions.
If a number is a multiple of two or more integers, it is called a common multiple. The smallest of the common multiples of a set of integers is called the the least common multiple (LCM).

How to Find LCD?

In order to find the LCD, firstly we convert all integers and mixed fractions into fractions. Then we find the LCM of the denominators. The result is the LCD and each fraction should be written as an equivalent fraction with the same LCD.
In some problems it is required to evaluate algebraic expressions. These problems can be solved by finding the least common denominator of two or more rational expressions fractions. For example, the least common denominator of rational expressions fractions $\frac{1}{4xy}$ and $\frac{1}{2x^2y}$ is $4x^2y$.In general, the least common denominator can be a number, a variable or a combination of numbers and variables. To find LCM two or more numbers, please refer to LCM calculator. Here, we will show how to find the LCD of $\frac{4}{5},\frac{6}{8},\frac{13}{11},\frac{2}{10}$ using the prime factorization. Therefore, it is necessary to find the LCM(5, 8, 11, 10).
Since every integer greater than 1 can be factored uniquely into primes, we obtain:
$$\begin{align}&5=5={\color{blue}5^1}\\ &8=2\times 2\times 2={\color{blue}{2^3}}\\ &11=11={\color{blue}{11^1}}\\ &10=2\times 5={2^1\times\color{blue}{5^1}}\end{align}$$ By multiplying the greatest power of 2, 11 and 5 form these factorizations, the LCM of 5, 8, 11, 10 is equal to the LCM of $\frac{4}{5},\frac{6}{8},\frac{13}{11},\frac{2}{10}.$ It means,

If we rewrite the original inputs as equivalent fractions using the LCD, we obtain the fractions $$\frac{352}{440}, \frac{330}{440}, \frac{520}{440}, \frac{88}{440}$$ The common denominator work with steps shows the complete step-by-step calculation for finding the least common denominator of a given set of numbers: $\frac{4}{5},\frac{6}{8},\frac{13}{11},\frac{2}{10}$ using the prime factorization. For any other set of numbers, just supply the list of numbers and click on the Generate Work button. The grade school students may use this common denominator to generate the work, verify the results of adding, subtracting, or comparing fractions, derived by hand or do their homework problems efficiently.

Real World Problems Using Least Common Denominator

In problems of adding, subtracting, or comparing fractions, we usually use the least common denominator. For instance, to compare rational numbers we use the LCD to rewrite them with a common denominator. As with fractions or mixed numbers, the LCD is very important in working with rational expressions. Many real life situations can be expressed by adding two or more rational expressions. So, the LCD can be useful here. Using the LCD is also important for solving some rational equations.

LCD Practice Problems

Practice Problem 1:
Find the common denominator of $\frac{1}{72}$ and $12\frac{1}{8}$.

Practice Problem 2:
Three persons, $A,B$ and $C$, paint a room together in 5 hours. The person $A$ paints the room alone in 9 hours. The person $B$ paints the room alone in 11 hours. Find the time needed for the person $C$ to paint the room.

The common denominator calculator, formula, step by step calculation, real world problems and practice problems would be very useful for grade school students (K-12 education) to learn how to find the LCD of two or more fractions, integers or mixed numbers. It can help in many math problems, particularly in adding, subtracting, or comparing fractions.