Fractions can be defined as numbers as can be represented in the form of A/B where A and B are integers and B should not be equal to zero. In a fraction, the upper part is called Numerator and the lower part is called the Denominator.

Examples: 1/2, 4/5, -2/3 etc.

Addition of Fractions

To add fractions there is a rule which states that theythat, then make them equal by taking the denominators’ Lowest Common Multiple(LCM)denominators of the fractions to be added should be equal. If the denominators of the fraction are not equal then make them equal by taking the Lowest Common Multiple(LCM) of the denominators.

How do I find LCM?

To find the LCM of numbers (here denominators), we will use the division I Method.

Let’s understand this method with the help of an example, take two numbers 6 and 15 for finding LCM using the division method.

LCM of 6,15

Step 1: Make a table that contains a left-hand side and a right-hand side, on the right-hand side put numbers whose LCM we are finding.

Step 2: Now start with the smallest number (not 1) and check if any number from the given numbers has this as its multiple. In the example, 2 is the factor of 6 so use it to divide 6 in the next row.

Step 3: Now in second row 3, 15 are left now, only factor of 3 is 3 so take 3 to divide it. 3 is also the factor of 15 so divide 15 also. The result is 1, 5.

Step 4: Now 5 is the factor of 5 so divide 5, the result is 1, 1.

Step 5: The process is complete as we get 1 for all numbers, now multiply all the numbers on the left-hand side which are 2,3,5 so multiple of these is 30.

Addition of 3 fractions with different denominators

Steps to add fractions with different denominators are:

Step 1: Find LCM of denominators.

Step 2: Divide the LCM by the denominator of each number which are to be added.

Step 3: Multiply the numerator with the quotient ( found in the above step).

Step 4: Add the numerators we get after multiplying with quotients like simple addition.

Step 5: The denominator will be the LCM.

Let’s take 3 fractions with different denominators, 1/2, 2/3, 3/4

Step 1: Finding LCM of 2,3,4

Step 2: Divide the LCM by the denominator of each number which are to be added.

LCM of 2,3,4

LCM = 12 so divide it by each number (denominator)

12/2 = 6 it is quotient 1

12/3 = 4 it is quotient 2

12/4 = 3 it is quotient 3

Step 3: Multiply the numerator with the quotient (found in the above step).

Numerators are 1, 2, 3 so multiply these with respective quotients.

1×6 = 6

2×4 = 8

3×3 = 9

Step 4: Add the numerators we get after multiplying with quotients like simple addition.

6 + 8 + 9 = 23 which is the numerator.

Step 5: Denominator will be the LCM so it is 12.

Answer is 23/12

Cross Multiplication Method

Taking the above example again, so add 1/2, 2/3, 3/4

Step 1: Take two fractions at a time so take 1/2 and 2/3

Step 2: First we will find the numerator terms so we multiply the numerator of the first number with the denominator of the second number and similarly we will multiply the numerator of the second number with the denominator of the first number and add both the terms to get numerator.

1×3 + 2×2 = 7 which is numerator

Step 3: Now let’s find the denominator, for this multiply the denominator of the first term with the denominator of the second term to get the denominator term.

2×3 = 6 which is the denominator.

Step 4: We find the new term which is the addition of two fractions in this case new fraction is 7/6.

Step 5: Repeat the above procedure taking the new fraction which is 7/6 and the third fraction which is 3/4.

LCM using Cross Method

Finally, we got the answer which is the same as found above. 

Sample Questions

Question 1: Add the given fractions, 1/7, 2/7, 3/7.

Answer: 

In the given question the denominators are equal so simply add the numerators and the denominator will be 7.

Adding numerators 1+2+3 = 6

Denominator = 7

Answer = 6/7.

Question 2: Find the LCM of 7, 3, 12.

Answer: 

LCM of 7,3,12

Question 3: Add the given fractions, 2/7, 5/12, 1/3.

Answer:

Step 1: Finding LCM of 7,12,3

LCM we got is 84.

Step 2: Divide the LCM by the denominator of each number which are to be added.

LCM = 84 so divide it by each number (denominator)

84/7 = 12 it is quotient 1

84/12 = 7 it is quotient 2

84/4 = 21 it is quotient 3

Step 3: Multiply the numerator with the quotient ( found in the above step).

Numerator are 2, 5, 1 so multiply these with respective quotients.

2×12 = 24

5×7 = 35

1×21 = 21

Step 4: Add the numerators we get after multiplying with quotients like simple addition.

24 + 35 + 21 = 80 which is the numerator.

Step 5: Denominator will be the LCM so it is 84.

Answer is 80/84

Question 4: Add the given fractions, 4/5, 3/10, 1/3.

Answer:

Step 1: Finding LCM of 5,10,3

LCM we got is 30.

Step 2: Divide the LCM by the denominator of each number which are to be added.

LCM = 30 so divide it by each number (denominator)

30/5 = 6 it is quotient 1

30/10 = 3 it is quotient 2

30/3 = 10 it is quotient 3

Step 3: Multiply the numerator with the quotient ( found in the above step).

Numerators are 4, 3, 1 so multiply these with respective quotients.

4×6 = 24

3×3 = 9

1×10 = 10

Step 4: Add the numerators we get after multiplying with quotients like simple addition.

24 + 9 + 10 = 43 which is the numerator.

Step 5: Denominator will be the LCM so it is 30.

Answer is 43/30

Question 5: Find the LCM of 7, 3, 12, 13

Answer:

LCM of 7,3,12,13

Question 6: Add the given fractions, 1/3, 1/4, 1/2 by cross multiplication method.

Answer: 

Step 1: Take two fractions at a time so take 1/3 and 1/4

Step 2: First we will find the numerator terms so we multiply the numerator of the first number with the denominator of the second number and similarly we will multiply the numerator of the second number with the denominator of the first number and add both the terms to get the numerator.

1×4 + 1×3 = 7 which is numerator

Step 3: Now let’s find the denominator, for this multiply the denominator of the first term with the denominator of the second term to get the denominator term.

3×4 = 12 which is the denominator.

Step 4: We find the new term which is the addition of two fractions in this case new fraction is 7/12.

Step 5: Again taking 7/12 and third fraction which is 1/2.

Step 6: Finding numerator

7×2 + 1×12 = 26 which is numerator

Step 7: Finding denominator

12×2 = 24 which is the denominator

The answer is 26/24 now simplifying it we will get 13/12.

Question 7: Add the given fractions, 1/5, 2/5, 3/10 by cross multiplication method.

Answer:

Step 1: Take two fractions at a time so take 1/5 and 2/5

Step 2: First we will find the numerator terms so we multiply the numerator of the first number with the denominator of the second number and similarly we will multiply the numerator of the second number with the denominator of the first number and add both the terms to get the numerator.

1×5 + 2×5 = 15 which is numerator

Step 3: Now let’s find the denominator, for this multiply the denominator of the first term with the denominator of the second term to get the denominator term.

5×5 = 25 which is the denominator.

Step 4: We find the new term which is the addition of two fractions in this case new fraction is 15/25 on dividing numerator and denominator by 5 we get 3/5.

Step 5: Now take 3/5 and the third fraction which is 3/10.

Step 6: Finding numerator

3×10 + 3×5 = 45 which is numerator

Step 7: Finding denominator

10×5 = 50 which is the denominator

The answer is 45/50 on dividing numerator and denominator by 5 we get 9/10.

Summary/Conclusion

In conclusion, adding fractions requires either making the denominators equal using the LCM method or using the cross-multiplication method. The LCM method involves finding the common denominator, adjusting the numerators accordingly, and then adding them. The cross-multiplication method involves pairing fractions, calculating new numerators and denominators step-by-step, and simplifying the result.

How to Add 3 Fractions with Different Denominators – FAQs

What is a fraction?

A fraction is a number that can be represented as A/B, where A and B are integers, and B is not zero. The upper part is called the numerator, and the lower part is the denominator.

How do you add fractions with different denominators?

To add fractions with different denominators, find the LCM of the denominators, adjust the numerators accordingly, and then add the numerators.

What is the LCM, and how do you find it?

LCM stands for Lowest Common Multiple. It is found by dividing the numbers using their smallest prime factors until all results are 1, and then multiplying the prime factors.

What is the cross-multiplication method for adding fractions?

The cross-multiplication method involves multiplying the numerator of each fraction by the denominator of the other, adding the results for the new numerator, and multiplying the denominators for the new denominator.

Can the cross-multiplication method be used for more than two fractions?

Yes, by pairing fractions, calculating new fractions step-by-step, and repeating the process.

Why is it necessary to find the LCM of denominators?

Finding the LCM of denominators ensures the fractions have a common denominator, making it possible to add their numerators directly.

What happens if the denominators of fractions are already equal?

If the denominators are already equal, you simply add the numerators and keep the common denominator.

Can these methods be used for subtracting fractions?

Yes, both the LCM and cross-multiplication methods can be adapted for subtracting fractions by subtracting the numerators instead of adding them.