Adding and Subtracting Fractions

multiples, common multiple, common denominator, fractions

Adding and Subtracting fractions has a methodical approach.  However, because most students do not know the correct method, they often stumble in mastering this art.  Here, we highlight a consistent method with which one can easily add and subtract fractions.

There are two cases to consider

I Case 1,

When denominators are the same:

when the denominators of the fraction we want to add is the same, we simply proceed to adding and or subtracting the numerators.

i.e. (5/7)-(1/7)=(4/7)

II Case 2,

When denominators are different:

When the denominators are different, we have a long process that we highlight in the following.

[1] Find a common denominator:

To find a common denominator, we can:

(a)Take the product of the two denominators of the fractions we have. This will not be the lowest common denominator if the two denominators have a common factor. It is, however, still usable, we will just need to simplify later on.

(b) We can list the multiples of the two denominators in two columns, then we can compare the two columns looking for a common multiple. The lowest number that is in both columns will be the lowest common denominator.

Either of the results of (a) and (b) will serve as our common denominator.


[2] Once we have a common denominator, we first consider how many times the common denominator goes into the denominator on the left and then multiply the numerator on the left by this result.

Then, we keep the addition or subtraction sign that appears in the original problem and we then consider

how many times the denominator on the right goes into the common denominator.  We then multiply this result to the numerator on the right.


[3] From here it is simple Arithmetic.  We do the math and we are done.


Example: if we want to do (7/4)-(5/9) we can either

[a] Multiply 4 and 9 to get 36 as a common denominator since 7 and 4 do not divide each other, or

[b] Find a common denominator by writing columns

i.e.  Multiples of 4 and 9 in two columns as follows:

4     9
8     18
12   27
16   36
20   45
24   54
28   63
32   72
36   81

Now when we compare the two columns, we see that 36 is a common multiple, and it is the smallest, so it is our lowest common denominator.  

With either method, we get that 36 is the common denominator, so now we write


  (7/4) – (5/9)
  ––––––––––
          36

and (a) consider how many times 4 [the denominator on the left] goes into 36, and we see that it is 9, and we take this result and multiply it to the numerator on the left, which is 7, to get:
                                  (9)(7)
                                   ––––––––––
                                           36

Then, we keep the subtraction sign and go to the right and (b) consider how many times 9 [the denominator on the right] goes into 36, and we see that it is 4, and we take this result and multiply it to the numerator on the right, which is 5, to get:
                                   (9)(7) – (4)(5)
                                   –––––––––––
                                            36


[3] We do the Math, from here, its simple and clear, we will just do the Math and get that 9*7=63 minus 4*5 which is 20 gives us 43, thus leaving us with:

                                          43
                                       –––––
                                          36

Our Final answer!!

This method is the way to tackle adding and subtracting fractions and works every single time.

multiples, common multiple, common denominator, fractions