7. Addition and Subtraction of Fractions
Recall that to add or subtract fractions, we need to have the same denominator.
Example 1
Add the fractions: `2/3+4/15`
Solution: Here, the lowest common denominator which we can use is `15`. So we have:
`2/3+4/15=10/15+4/15=14/15`
Subtraction works in the same way, as we see in the next example.
Example 2
`6/7-5/14`
Answer
Our lowest common denominator this time is 14. So we have:
`6/7-5/14=12/14-5/14=7/14=1/2`
When we have algebraic expressions involving fractions, we need to use the same process.
Example 3
`a/(6y)-(2b)/(3y^4)`
Answer
The lowest common denominator here will be 6y4.
So we have:
`a/(6y)-(2b)/(3y^4)=(ay^3)/(6y^4)-(4b)/(6y^4)`
We can write this as:
`(ay^3-4b)/(6y^4`
Example 4
`(x-1)/(2x^3-4x^2)+5/(x-2)`
Answer
We factor the first denominator to get an idea of what to do:
`(x-1)/(2x^3-4x^2)+5/(x-2)`
`=(x-1)/(2x^2(x-2))+5/(x-2`
We can see now that if we multiply the second fraction by 2x2, we will have a common denominator:
`(x-1)/(2x^2(x-2))+(5xx2x^2)/(2x^2(x-2))`
`=(x-1+10x^2)/(2x^2(x-2)`
We would normally write this as:
`(10x^2+x-1)/(2x^2(x-2)`
Exercises
Simplify the following.
(1) `2/(s^2)+3/s`
Answer
`2/(s^2)+3/s`
`=2/(s^2)+(3s)/(s^2)`
`=(2+3s)/(s^2)`
(2) `5+(1-x)/2-(3+x)/4`
Answer
The lowest common denominator is 4.
`5+(1-x)/2-(3+x)/4`
`=20/4+(2(1-x))/4-(3+x)/4`
`=(20+2-2x-3-x)/4`
`=(-3x+19)/4`
(3) `5/(6y+3)-a/(8y+4)`
Answer
`(5)/(6y+3)-(a)/(8y+4)` `=5/(3(2y+1))-a/(4(2y+1))`
`=20/(12(2y+1))-(3a)/(12(2y+1))`
`=(20-3a)/(12(2y+1))`
We have factored out the 3 at the bottom of the first fraction and the 4 at the bottom of the second fraction.
In the 3rd line, we find the lowest common denominator, 12(2y + 1), and multiply top and bottom of the two fractions accordingly.
The last line is a tidy up step.
