Adding and Subtracting Fractions

If two fractions have the same denominators, we can easily perform the addition or subtraction. For addition, simply add the numbers on top. The denominator remains the same.


Example

\frac { 4 }{ 7 } +\frac { 2 }{ 7 } =\frac { 4+2 }{ 7 } =\frac { 6 }{ 7 }
To subtract, we subtract the second number from the first. The denominator remains the same
.


Example

\frac { 4 }{ 7 } -\frac { 3 }{ 7 } =\frac { 4-2 }{ 7 } =\frac { 1 }{ 7 }
If the fractions differ in their denominators, we multiply the top and bottom by appropriate numbers to make them equal, before we add or subtract.


Example:

\frac { 5 }{ 12 } +\frac { 3 }{ 4 }
Then we convert \frac { 3 }{ 4 } as \frac { 3\times 3 }{ 4\times 4 }, multiplying the top and bottom by 3.
Now \frac { 3\times 3 }{ 4\times 3 } =\frac { 9 }{ 12 }
Therefore \frac { 5 }{ 12 } +\frac { 3 }{ 4 } =\frac { 5 }{ 12 } +\frac { 9 }{ 12 } =\frac { 5+9 }{ 12 } =\frac { 14 }{ 12 }
Which simplifies to \frac { 7 }{ 6 } or 1\frac { 1 }{ 6 }


Example:


Find \frac { 5 }{ 8 } -\frac { 2 }{ 7 }
We convert \frac { 5 }{ 8 } and \frac { 2 }{ 7 } as \frac { 5\times 7 }{ 8\times 7 } and \frac { 2\times 8 }{ 7\times 8 }
Or \frac { 35 }{ 56 } and \frac { 16 }{ 56 }. Then we do the subtraction:


\frac { 5 }{ 8 } -\frac { 2 }{ 7 } =\frac { 35 }{ 56 } -\frac { 16 }{ 56 } =\frac { 19 }{ 56 }


A much harder example is the following:



Find \frac { 2 }{ 7 } +\frac { 2 }{ 5 } -\frac { 1 }{ 8 }

We should first express the given fractions with the same denominator. To do this, we need what we call the least common multiple of the numbers7,5 and 8.
The lease common multiple (LCM)



First express the numbers 7,5 and 8 in their prime factorisation.
7=7
5=5
8=2\times 2\times 2
To find the LCM, choose the smallest multiple common to the numbers. Here this is 2\times 2\times 2\times 5\times 7=280
So, we can rewrite the fractions \frac { 2 }{ 7 } ,\frac { 3 }{ 5 },and \frac { 1 }{ 8 } with the denominator 280.

  • \frac { 2 }{ 7 } =\frac { 2 }{ 7 } \times \frac { 40 }{ 40 } =\frac { 80 }{ 280 }
  • \frac { 3 }{ 5 } =\frac { 3 }{ 5 } \times \frac { 56 }{ 56 } =\frac { 168 }{ 280 }
  • and \frac { 1 }{ 8 } =\frac { 1 }{ 8 } \times \frac { 35 }{ 35 } =\frac { 35 }{ 280 }
  • Therefore \frac { 2 }{ 7 } +\frac { 3 }{ 5 } -\frac { 1 }{ 8 } =\frac { 80 }{ 280 } +\frac { 162 }{ 280 } -\frac { 35 }{ 280 }
  • =\frac { (80+168-35 }{ 280 }
  • =\frac { 213 }{ 280 }


We consider another example


Find \frac { 3 }{ 16 } +\frac { 5 }{ 24 } -\frac { 1 }{ 30 }, giving your answer in its simplest form.
16=2\times 2\times 2\times 2
24=2\times 2\times 2\times 3
30=2\times 3\times 5
So 2\times 2\times 2\times 2\times 3\times 5 is the LCM
2\times 2\times 2\times 2\times 3\times 5=240

  • Now \frac { 3 }{ 16 } =\frac { 3 }{ 16 } \times \frac { 15 }{ 15 } =\frac { 45 }{ 240 }
  • \frac { 5 }{ 24 } =\frac { 5 }{ 24 } \times \frac { 10 }{ 10 } =\frac { 50 }{ 240 }
  • and \frac { 1 }{ 30 } =\frac { 1 }{ 30 } \times \frac { 8 }{ 8 } =\frac { 8 }{ 240 }
  • Hence \frac { 3 }{ 16 } +\frac { \cfrac { 5 }{ 2401 } }{ 30 } =\frac { 45+50-8 }{ 240 }
  • =\frac { 87 }{ 240 }
  • Since 3 is common from the top 87 and the bottom 240, we can remove it by dividing the top and bottom by 3.
  • \frac { 87 }{ 240 } =\frac { 29\times 3 }{ 80\times 3 }
  • =\frac { 27 }{ 80 }


29 is the prime, so that 29/80 is the simplified form.



Evaluate

=\frac { 24 }{ 25 } -\frac { 1 }{ 75 }
25=5\times 5
75=3\times 5\times 5
LCM(25,75)=3\times 5\times 5=75

  • \frac { 24 }{ 25 } =\frac { 24 }{ 25 } \times \frac { 3 }{ 3 } =\frac { 72 }{ 75 } and \frac { 1 }{ 75 } =\frac { 1 }{ 75 }.
  • Therefore \frac { 24 }{ 25 } -\frac { 1 }{ 75 } =\frac { 72 }{ 75 } -\frac { 1 }{ 75 }
  • =\frac { 71 }{ 75 }
  • 71 is a prime number, so we cannot divide it anymore. \frac { 71 }{ 75 } is the simplest form.