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How to Add Fractions with Unlike Denominators?

  • Last Updated : 18 Jan, 2022

Fractions are the numbers in the form of a/b where a and b are whole numbers where b ≠ 0. Some examples of fractions are 2/5, 3/7, 3/1, 6, etc. The whole numbers such as 6 are also fractions since 6 can be written as 6/1. 

Properties:

  • For a fraction, a and b, both a and b are whole numbers with b ≠ 0.
  • All whole numbers are fractions with a denominator equals 1
  • Fractions can be added, subtracted, multiplied, and divided with each other.

Addition of Fractions

The methods to add fractions are:

  • Cross multiplication: Cross multiplying the two fractions and adding the numerator parts of both the fractions to get the numerator of the answer. The denominator parts can be multiplied by each other to get the final denominator of the answer. Since they may not be in their reduced form, we may need to divide the numerator and denominator by a common factor to get a proper fraction.

Consider two fractions a/b and c/d. Then, their sum can be done by:

a/b + c/d = (ad + bc) / bd

For example,

 3/5 + 2/7 = (7×3 + 2×5) / 5×7 = (21 + 10) / 35 = 31/35

  • LCM of the denominator: In this method, we need to find the LCM of all the denominators of the fractions and then multiply the same factors with both numerator and denominator of each fraction so that denominator is the same for all the fractions. Then we need to add all the new numerators of the fractions to get the numerator of the answer. The LCM of all the denominators becomes the denominator of the answer.

Consider two fractions 3/8 and 2/3. To add the fractions, we need to find the LCM of 3 and 8.

Now, LCM of 3 and 8 is 24. Thus, we proceed as:

3/8 + 2/3 = (3×3 / 8×3) + (2×8 / 3×8) = 9/24 + 16/24 = 25/24

Thus, the sum is equal to 25/24.

Sample Problems

Question 1. Find the sum of the fractions 2/5 and 3/4 using the Cross multiplication method.

Solution:

To apply the cross multiplication method, we find the sum of numerators by cross multiplying them with other denominator.

Thus, the numerator of the answer = 2×4 + 3×5 = 23

The denominator of the answer = product of the denominators = 5×4 = 20

Thus, the sum is equal to 23/20.

Question 2. Find the sum of the fractions 3/5 and 5/2 by using the LCM method.

Solution:

To apply the LCM method, we first need to find the LCM of the denominators 2 and 5. 

The LCM of 2 and 5 is 10.

Thus, we can write,

3/5 + 5/2 = 3×2 / 5×2 + 5×5 / 2×5 = 6/10 + 25/10

Thus, 3/5 + 5/2 = 31/10

Question 3. Find the sum of the fractions 3/5, 6/7, and 3/2.

Solution:

To find the sum of the three fractions, we need to first find the sum of the two fractions and add the answer to the third fraction. 

At first, we add 3/5 and 6/7 using the cross multiplication method. Thus,

3/5 + 6/7 = (3×7 + 6×5) / (5×7) = (21 + 30) / 35 = 51/35

Now, we add 51/35 with 3/2 using the same method to get out final answer.

51/35 + 3/2 = (51×2 + 35×3) / (35×2) = 207/70

Since 207/70 is a proper fraction and cannot be reduced further, the answer is 207/70.

Question 4. Find the fraction to be added with 5/6 to get a sum equal to 3/2.

Solution:

Let the fraction to be added be x. Then we can write

x + 5/6 = 3/2

x = 3/2 – 5/6

We can use the cross multiplication method to find the numerator of x and the denominator can be found by finding the product of both the denominators.

Thus, we can write

x = (3×6 – 5×2) / (2×6)

x = (18 – 10) / 12

x = 8/12

Since 8/12 is not in reduced form and can be further reduced by dividing both the numerator and denominator with their HCF.

HCF of 8 and 12 = 4

Thus, we can write

8/12 = (4×2) / (4×3) = 2/3

Thus, x = 8/12 = 2/3

So, 2/3 should be added with 5/6 to get a sum equal to 3/2.

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