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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

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)

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)

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

2/(s^2)+3/s

=2/(s^2)+(3s)/(s^2)

=(2+3s)/(s^2)

(2) 5+(1-x)/2-(3+x)/4

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)

(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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