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Adding and Subtracting Fractions - with Different Denominators

 

To add or subtract fractions with different denominators, we need to do some extra steps. The general approach is discussed below. We will go over few examples in this lesson to make sure you get comfortable with the procedure.

 

Steps to add/subtract fractions with different denominators
Step 1: Given two unlike fractions where b is not equal to d. fraction a/b and fraction c/d

 

Step 2: Make the denominators the same by finding the Least Common Multiple (LCM) of their denominators.

 

LCM(b,d)=m
Step 3: Rewrite each fraction into its equivalent fraction with a denominator of LCM of (b,d) equals m.

 

a/b implies a'/m, and c/d implies c'/m

 

Step 4: Now, add or subtract the "new" fractions from step 3.

  • Always reduce the answer to its lowest terms.

 

(a'/m)+(c'/m)=(a'+c')/m, and (a'/m)-(c'/m)=(a'-c')/m

 

 


Example 1: Add the fractions with different denominators sum of (2/15) and (3/5), or (2/15)+(3/5).

The two fractions have denominators that are not equal. We need to make them equal by finding their Least Common Multiple that will serve as their Least Common Denominator (LCD). 

Start by listing the multiples of each denominator, and identify the least number that is common to both of them.

 

  multiples of each denominator common denominator
multiples of fifteen or 15 multiples of 15 are 15, 30, 45, 60, and so on LCM(15,5)=15
multiples of five or 5 multiples of 5 are 5, 10, 15, 20, 25, and so on

 

The first fraction already has a denominator equal to the LCM = 15, and so we will leave it alone.

The second fraction requires some adjusting to make its denominator equal to 15. Do that by multiplying its numerator and denominator by the number 3.

  • Once their denominators are equal, add the fractions by adding their numerators and then copying the common denominator.
sum of fractions: (2/15)+(3/5)=11/15

The fraction eleven over 15 (11/15) is our final answer because it is already in its lowest term.


Example 2: Add the fractions with different denominators sum of fractions (2/5) and (3/9), or simply (2/5)+(3/9).

We can't add the two fractions just yet because they have different denominators, namely 5 and 9. Begin by listing their multiples and pick the smallest number that is common to both. This will become their common denominator.

 

  multiples of each denominator common denominator
multiples of five or 5 multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50,55, 60, and so on. LCM(5,9)=45, or the least common multiple of 5 and 9 is 45
multiples of nine or 9 the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, and so on

 

Now, convert each fraction to an equivalent fraction with the LCM as its denominator, then proceed with regular addition.

Look for the opportunity to reduce the answer to its lowest term. The numerator and denominator of 33/45 is divisible by 3.

add fractions 2/5 and 3/9 which is (2/5)+(3/9)=11/15

 


Example 3: Add the fractions with different denominators add fractions (2/3)+(1/5).

Sometimes there is no "need" to find the least common denominator by list method. We can immediately find it whenever the two numbers are both prime.

  • A prime number is a number divisible only by 1 and itself.

Observe that the denominators 3 and 5 are primes. The LCD will simply be their product, that is, 3 times 5 equals 15.

add fractions: (2/3)+(1/5)=13/15

 


Example 4: Add the fractions with different denominators sum of two fractions: (6/18)+(3/6).

Solution:

Find the least common multiple of the the denominators.

  multiples of each denominator common denominator
multiples of eighteen multiples of 18 are 18,36,54,72 , etc LCM(18,6)=18
multiples of six multiples of 6 are 6,12,18,24,30,36, etc

 

Make the necessary adjustments in the denominator and proceed as usual. Reduce your final answer to the lowest term.

add fractions: (6/18) + (3/6) = 5/6

 


Example 5: Add the fractions with different denominators sum of fractions: (1/11)+(2/13).

Solution:

Since the denominators eleven and thirteen are both prime numbers, the least common denominator will be their product.

 

  • LCD = 11 x 13 = 143

 

Convert the current denominators of the two fractions into the LCD, and proceed with regular addition.

adding two fractions: (1/11)+(2/13) = 35/143

 

 


Example 6: Subtract the fractions with different denominators subtract fractions: (9/10)-(5/6).

To subtract these fractions with unequal denominators is very similar to addition.

Make their denominators equal using the concept of least common multiple. Then subtract their numerators accordingly.

 

  multiples of each denominator common denominator
multiples of ten multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, etc LCM of  10 and 6 equals 30
multiples of six multiples of 6 are 6,12,18,24,30,36,42,48,54,60,66, etc

 

Rewrite each fraction to its equivalent fraction with a denominator equal to the LCM = 30, then subtract their numerators. Make sure to reduce your answer to the lowest term.

 

subtract fractions: (9/10) - (5/6) = 1/15

 


Example 7: Subtract the fractions with different denominators subtract fractions: (6/7)-(3/5).

Since the denominators are both prime numbers, their LCM is just their product, thus seven times five equals 35.

subtract fractions: (6/7)-(3/5)=9/35

 


Example 8: Subtract the fractions with different denominators subtract fractions (2/14) - (5/6).

Solution:

Find the least common denominator by determining the LCM of the denominators.

 

  multiples of each denominator common denominator
multiples of fourteen multiples of 14 are 14, 28, 42, 56,70, etc LCM of 14 and 6 equals 42
multiples of six multiples of 6 are 6,12,18,24,30,36,42,48,54, etc

 

Rewrite the two fractions with a common denominator equal to the LCM = 42. Subtract their numerators, and reduce the answer to the lowest term if possible.

subtracting fractions: (2/14)-(5/6)=-29/42

 

 


Example 9: Subtract the fractions with different denominators subtract 1/8 and 1/20.

Solution:

Find the least common denominator by solving for the least common multiple of the denominators.

  multiples of each denominator common denominator
multiples of eight multiples of 8 are 8,16,24,32,40,48,56, etc LCM (8,20) = 40
multiples of twenty multiples of 20 are 20, 40, 60, 80, 90, etc

We make adjustments on the existing fractions to make their denominator equal to the LCD = 40. After doing so, subtract their numerators and copy the common denominator.

subtracting fractions: (1/8) - (1/20) = 3/40

 


 

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