A common denominator is useful for conducting a number of mathematical operations on numbers. A crucial concept in Mathematics is the addition and subtraction of fractions. Fractions or a fractional number has 2 parts that include a numerator (the number on top) and a denominator (the number below). That said, the fractions with the same denominators are called common denominators. Consider the following denominator examples: 3/7 + 5/7 = 3. In these cases, the denominators in the fraction are common, thus, it is also easy to calculate the answer.
Methods to Find a Common Denominator?
Want to know how to find the common denominator easily? There may be instances where you would be asked to add fractions with different denominators like 5/7 + 9/13. In such times, you are required to determine the common denominator and then solve the fractions. Following are the two methods to find the common denominator:
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By cross multiplication
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By finding the least common multiple (LCM)
When we solve for the common denominator using the LCM (least common multiple) method, you find the LCM of the given numbers. In this equation, the LCM is 6. Thus, the equation becomes 1/3 + 1/6 = (1 x 2 + 1)/6 = (2 + 1)/6 = 3/6 = 1/2. However, If you the cross multiplication method, you will find the solution as: 1/3 + 1/6 = 2/6 + 1/6 = (2 + 1)/6 = 3/6 = 1/2
How to Find the Least Common Denominator?
The least common denominator depends upon the type of denominator. For denominators having co-prime numbers, the least common denominator is the product of the two denominators. In addition, the least common denominator is the LCM of the two given denominators. Having said that, let us take two denominator values: 6 and 4. As said, the least common denominator is the least common multiple of 4 and 6, which is the number 12.
Now, taking the case of numbers 3 and 4. In such an instance, neither 3 nor 4 are factors of each other. Therefore, you can calculate the value of the common denominator simply by multiplying both numbers and you obtain 12.
What is the Greatest Common Denominator?
In mathematics, the Greatest Common Denominator of two or more fractions, which are not zero (0), is the biggest positive integer which divides each of the given denominators.
Can a Common Denominator be 0 or 1?
For a fraction having a common denominator of zero (0), it becomes undefined. Whereas, for fractions with only whole numbers as numerators and 1 as a denominator, the common denominator will be 1. In the instance of whole numbers being taken into account as fractions, the common denominator is 1.
Fractions Without a Common Denominator
We can get the common denominators by multiplying both numerator (the top number) and denominator (the bottom number) by the same amount.
For example, take out the addition of two fractions which do not have a similar denominator:
Addition of 2/5 and 1/2.
Firstly, multiply 2/5 by 2/2 to obtain 4/10 .
Now, multiply the 2nd fraction 1/2 by 5/5 to obtain 5/10 .
The new fractions with a common denominator are 10.
Add the two fractions 4/10 + 5/10 , you will get 9/10.
Solved Examples on Common Denominator
Example:
Can you help Alex find the common denominator for the fractions 7/5 and 4/3?
Solution:
The denominators of the given fractions are 5 and 3 respectively. Thus, the LCM of 5 and 3 is 15 and 15 is the common denominator of the two given fractions.
Therefore, the common denominator for 7/5 and 4/3 2/3 is 15.
Example:
Find out if the rational numbers -9/12 and 21/-28 equal?
Solution:
The two given fractions can be simplified as:
-9/12 = -3/4
21/-28 = 3/-4 = -3/4.
Seeing that, the simplified value of the given two fractions is the same, thus, the two rational numbers are also equal.
∴ We can conclude that the two rational numbers -9/12 and 21/-28 are equal.
Fun Fact
The LCM of the denominators helps us in getting the common denominator.