In this section, we will learn how to add and subtract fractions.  

 

Fractions can be added or subtracted only when they have the same denominators. 

Adding and Subtracting Fractions with Same Denominators: 

ab+cb=a+cb

ab−cb=a−cb

For example: 15+35=45 ,          and              15−35=−25 .

 

Adding or subtracting fractions with different denominators requires us to first find the least common denominator (LCD).

Adding and Subtracting Fractions with Different Denominators: 

Step 1. Finding the Least Common Denominator (LCD)

  1.  Make a list of (enough) multiples of each denominator.
  2. Identify the lowest common multiple. If you can’t see one, then your lists need to be expanded.

Step 2.  Rewrite both fractions (by multiplying both numerator and denominator by the appropriate same number) to get the LCD as denominator.

Step 3. Add or subtract then simplify if needed. 

Example: Find the LCD and then add and simplify 312+58

Step 1.  Finding the LCD of 12 and 8

                 8:8,16,24,36,…

                 12:12,24,36,…

                  LCD:   24

Step 2.  Rewrite each fraction using the LCD

                 312=3⋅212⋅2=624 

                 58=5⋅38⋅3=1524 

Step 3.  Add or subtract then simplify if needed. 

                 624+1524=2124=3⋅73⋅8=78 

The above three steps can be combined as:

                 312+58=3⋅212⋅2+5⋅38⋅3=624+1524=2124=78 

 

Example: Find the LCD and then subtract and simplify 19−35

Step 1.  Finding the LCD of 9 and 5

                 9:9,18,27,36,45,54,…

                 5:5,10,15,20,25,30,35,40,45,50,…

                  LCD:   45

Step 2.  Rewrite each fraction using the LCD

                 19=1⋅59⋅5=545 

                 35=3⋅95⋅9=2745 

Step 3.  Add or subtract then simplify if needed. 

                 545−2745=−2245 

The above three steps can be combined as:

                 19−35=1⋅59⋅5−3⋅95⋅5=545−2745=−2245 

 

 

Practice Problems: 

1.   Add or subtract the following fractions:

(a)   35−56

(b)   −711+233 

(c)   13−12

(b)   −12+2    (note that 2=2/1)

 

Answer Key: (a)         (b)          (c)            (d)                       

 

For more detailed explanation, please read: Arithmetic|Algebra Chapter 2.