Definition of Like Fractions :
The fractions which are having the same denominator are called as like fractions.
Examples :
3/5, 6/5, 2/5, 7/5
In all the above fractions, the denominator is same. That is 5.
Definition of Unlike Fractions :
The fractions which are having different denominators are called as unlike fractions.
Examples :
3/5, 6/7, 2/9, 7/2
In the above fractions, denominators are different. They are 5, 7, 9 and 2.
The above said stuff is clearly illustrated in the picture shown below.
When students want to identify the like fractions and unlike fractions, they have to do a simple work.
That is, they just have to look at the denominators (bottom numbers) of the fractions.
If they are same, they can say that they are like fractions.
In case the denominators are different, they can say that they are unlike fractions.
We can understand the difference between like fractions and unlike fractions from the definitions explained above.
The difference between like fractions and unlike fractions will play a key role in adding and subtracting of fractions.
When we add two or more like fractions, we will take the denominator once and simplify the numerators.
Example :
2/7 + 4/7 = (2 + 4)/7
2/7 + 4/7 = 6/7
From the above example, it is clear that adding or subtracting two or more like fractions is easy.
But, when we add or subtract two ore more unlike fractions, we have to use different methods.
They are,
1. Cross-Multiplication method
2. LCM Method
Cross - Multiplication Method :
In addition or subtraction of two unlike fractions, if the denominators are co-prime or relatively prime, we have to apply this method.
We have to follow the steps explained below in cross multiplication method.
Step 1 :
Multiply the numerator of the first fraction by the denominator of the second fraction.
Step 2 :
Multiply the numerator of the second fraction by the denominator of the first fraction.
Step 3 :
Multiply the denominators of both fractions and take it as common denominator for the results of step 1 and step 2.
Step 4 :
Simplify the result of step 3.
Example :
Let us consider the addition of two fractions given below.
2/7 + 3/8
In the above two fractions, denominators are 7 and 8.
For 7 and 8, there is no common divisor other than 1.
So 7 and 8 are co-prime.
Here, we have to apply cross-multiplication method to add the two fractions 2/7 and 3/8 as shown below.
2/7 + 3/8 = [(2 ⋅ 8) + (3 ⋅ 7)]/(7 ⋅ 8)
2/7 + 3/8 = [16 + 21]/(56)
2/7 + 3/8 = 37/56
LCM (Least Common Multiple) Method :
In addition or subtraction of two unlike fractions, if the denominators of the fractions are not co-prime (there is a common divisor other than 1), we have to apply this method.
We have to follow the steps explained below in LCM method.
Step 1 :
Find the least common multiple of the denominators of the given fractions.
Step 2 :
Using the least common multiple found in step 1, make all the fractions as like fractions.
Step 3 :
In like fractions, the denominator of all the fractions will be same. So take the denominator once and simplify the numerators.
Example :
Let us consider the addition of two fractions given below.
3/8 + 5/12
In the above two fractions, denominators are 8 and 12.
For 8 and 12, if there is at least one common divisor other than 1, then 8 and 12 are not co-prime.
For 8 and 12, we have the following common divisors other than 1.
2 and 4
So, 8 and 12 are not co-prime.
In the next step, we have to find the LCM (Least common multiple) of 8 and 12.
8 = 2^{3}
12 = 2^{2} x 3
When we decompose 8 and 12 in to prime numbers, we find 2 and 3 as prime factors for 8 and 12.
To get L.C.M of 8 and 12, we have to take 2 and 3 with maximum powers found above.
So, the LCM of 8 and 12 is
= 2^{3} x 3
= 8 x 3
= 24
Now, make the denominators of both the fractions as 24 using multiplication.
In (3/8 + 5/12), to make each denominator as 24, multiply the numerator and denominator of the first fraction by 3 and the second by 2.
That is,
3/8 + 5/12 = 9/24 + 10/24
3/8 + 5/12 = (9 + 10)/24
3/8 + 5/12 = 19/24
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