Addition of Three Fractions with Different Denominators

What are Fractions?

Fractions, in simple terms, mean a part of a whole. Fractions in mathematics are represented by two numbers, one on the top and the other on the bottom, with a line between them. These numbers are called the numerator and the denominator. The numerator represents how many parts are present, and the denominator represents how many parts something is divided into.

For Example: Consider dividing a pizza into 4 equal halves. We call each part a fraction of a whole pizza. Here, the denominator is 4, the numerator is 4, and the fraction becomes 4/4, which is equal to 1 pizza.

Now imagine, if one piece of the pizza is eaten, then the denominator still remains 4 as the pizza was cut into 4 halves, and the numerator changes to 3. Now the fraction becomes ¾.

LCM

LCM is the short form of Least Common Multiple. When two or more numbers have the same number as their multiples, then that number is said to be the common multiple.

A Least Common Multiple of two or more numbers is the smallest number that is a multiple of those numbers.

Steps to determine LCM

LCM of the given numbers can be determined by the Common division method.

Step1:  Divide the given numbers by the smallest prime number, which divides at least one of the numbers given. Generally, we begin with the smallest prime number, which is 2. If the given number is not divisible by 2, then they are carried down to the next step.

Step2: Continue dividing the quotients by the least prime factor until the quotient becomes 1. We can consider the next prime number, which is 3, and so on.

Step3: The product of prime factors of all the numbers gives the LCM.

Example: LCM of 25, 15 and 30

Step1: Dividing the numbers by the smallest prime number that is 2:

30 is divisible by 2; thus, 30/2=15. Since 15 and 25 are not divisible by 2, they are carried on to the next step.

Step2: Dividing the numbers by the next smallest prime number, that is 3:

• From the previous step, we already have 15 as the quotient.
• Since 15 is divisible by 3, we divide them and carry forward the number 25 as it is not divisible by 3.
• Consider the next prime number, which is 5. Both, 25 and 5 are divisible by 5; thus, 25/5=5 and 5/5=1
• Since only 5 is remaining, we can divide it by 5.

Step3: Now multiply all the factors to obtain LCM.

Therefore LCM=2×3×5×5, which is the same as 2¹×3¹×5², is 150

Addition of fractions with different denominators

Step1: In order to add three fractions with different denominators, it is necessary that we have a common denominator for all the fractions. To obtain a common denominator, we need to find the LCM of the denominators of all the fractions.

Step2: Once the common denominator is calculated, each fraction is then multiplied by this common denominator.

Step3: The number obtained by doing this for each fraction is added to the numerator.

Step4: Now, we obtain both the numerator and the denominator (which is the LCM), which can be further simplified.

Examples:

(2/5) *15 =6, (2/3) *15=10 and (3/15) *15 =3