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Adding and Subtracting Fractions - Same or Like Denominators

 

When you add or subtract fractions, consider the problem to be easy when the denominators are equal or the same. The rules can be summarized below.

 

Fractions with the same denominators

To ADD

To SUBTRACT

 

 

formula how to add fractions: (a/d)+(b/d)=(a+b)/d

 

 

 

formula to subtraction fractions: (a/d)-(b/d)=(a-b)/d where d is the common denominator

 

Comment: To add fractions with equal denominators, simply add the numerators then copy the common denominator. Comment: To subtract fractions with equal denominators, simply subtract the numerators then copy the common denominator.

 


Example 1: Add the fractions add fractions: (3/7)+(2/7).

The denominators of the two fractions are both 7. By having the same denominators, we can easily add these fractions by adding their numerators and copying the common denominator which is 7.

(3/7)+(2/7)=5/7

We can also show the addition process using circles.

  • The first fraction 3/7 can be represented by a circle divided equally in seven parts with three pieces shaded in red.

Observe: the numerator tells us how many areas are shaded while the denominator tells us how many equal parts the circle is divided.

pie chart of fraction 3/7
  • In the same manner, the second fraction 2/7 looks like this.
pie chart representation of fraction 2/7

 

  • Since the two circles are both divided in seven (7) equal parts, we should be able to overlap them. The new circle after addition has five (5) shaded regions which is the accumulation of both red and blue pieces.

 

pie charts showing how to add (3/7) and (2/7) which is (3/7)+(2/7)=5/7

 


Example 2: Add the fractions add fractions: (3/16)+(9/16).

Let's combine these fractions using the addition rule. Again, add the numerators and copy the common denominator.

(3/16)+(9/16)=12/16

After you add fractions, always find the opportunity to simplify the added fractions by reducing it to the lowest term. We can do so by dividing both the numerator and denominator by their greatest common divisor.

  • common divisor is a nonzero whole number that can evenly divide two or more numbers.
  • greatest common divisor (GCD) is the largest number among the common divisors of two or more numbers.

Obviously, the numerator and denominator have a common divisor of 2.  However, is there a number larger than 2 that can also evenly divide both of them?

Yes, there is! The number 4 is the greatest common divisor of 12 and 16. Therefore, we will use this number to reduce the fraction to its lowest term.

Divide the top and bottom by the GCD = 4 to get the final answer.

 

(3/16)+(9/16)=12/16=3/4 when simplified to lowest terms

 


Example 3: Add the fractions add fractions: (5/30) + (1/30).

Solution:

Since the denominators of the two fractions are equal, add the numerators and copy the common denominator.

(5/30)+(1/30)=6/30, not simplified to lowest terms

The top and bottom numbers of the fraction are divisible by 2 and 6. However, we always want the largest common divisor to reduce the fraction to its lowest term. Thus, the GCD = 6.

  • Divide the top and bottom numbers by 6.
(5/30)+(1/30)=6/30=1/5, answer in lowest terms

 


Example 4: Add the fractions add the three fractions: (3/45)+(14/45)+(8/45).

Solution:

All three fractions have the same denominators. The rule in adding fractions with equal denominators still holds!

  • Get the sum of the three numerators, and copy the common denominator.
(3/45)+(14/45)+(8/45)=25/45, not yet reduced to lowest terms

The greatest common division between the numerator and denominator is 5.

  • Divide top and bottom by 5.
(3/45)+(14/45)+(8/45)=25/45=5/9, reduced to lowest term

 


Example 5: Subtract the fractions subtract fractions: (5/5)-(2/5) .

This time around, we are going to subtract the numerators instead of adding them.

(5/5)-(2/5)=3/5

Looking at the result after subtraction, the only common divisor between the numerator and denominator is 1. Thus, the final answer remains to be 3/5. Think about it, dividing the top and bottom by 1 won't change the value of the fraction.

How does it look graphically?

Suppose you have a green cake. And you cut it in 5 equal portions. This can be represented in fraction as 5/5.

pie graph of 5/5 (circle divided into five equal parts)

If you ate two slices of the cake ( -2/5 ) , you should have three leftover pieces ( 3/5 ).

The plate should look something like this.

 

pie chart of the fraction 3/5

 


Example 6: Subtract the fractions subtract fractions: (10/27)-(4/27) .

The two fractions have the same denominators which means we should be able to easily subtract their numerators.

(10/27)-(4/27)=6/27, not simplified yet

The answer can still be further simplified using a common divisor of 3. So, divide the numerator and denominator by 3 to reduce the fraction to its lowest terms.

(10/27)-(4/27)=6/27=2/9, now in simplest form

Example 7: Subtract the fractions subtract fractions: (21/81)-(3/81).

Solution:

Since the denominators of the two fractions are equal, subtract their numerators and copy the common denominator.

(21/81)-(3/81)=18/81, not in simplest term

The numerator and divisor are divisible by 3 and 9. However, we always want the largest common divisor to reduce the fraction to its lowest term. Thus, the GCD = 9.

  • Divide the top and bottom numbers by 9.
(21/81)-(3/81)=18/81=2/9, in simplest form

 


Example 8: Subtract the fractions subtract three fractions: (22/33)-(6/33)-(5/33)=11/33=1/3, final answer in simplest or reduced form.

Solution:

Subtract the numerators, and reduce the resulting fraction to its lowest term using the GCD = 11.

(22/33) - (6/33) - (5/33) = 1/3

 


 

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About This Topic: Add/Subtract Fractions with the Same Denominators

 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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