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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`

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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)`

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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)`

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(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`

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(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.

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