How is adding, subtracting, multiplying and dividing fractions similar to doing

Question

How is adding, subtracting, multiplying and dividing fractions similar to doing the same with rational expressions. can you give me an example of each?

Answer 1

Suppose I want to add 1/4 + 2/4. Since they have common denominators we can add the numerators to give 3/4. 

If the denominators are not the same we must first find (preferably) the lowest common denominator (though any common denominator will suffice).

2/3 + 3/5 = ?/15

We can say, 3 goes into 15 5 times, and 5 times 2 is 10. And 5 goes into 15 3 times, and 3 times 3 is 9. This would give (10 + 9)/15 = 19/15.

What about 3/5 + 3/10? The LCD is 10, however, we could also use (5)*(10) since a common denominator can always be found by taking the product of the denominators. Instead of multiplying the denominator, let's keep it as a product of the two factors: 

____?____
 (5)*(10)

We can say, 5 goes into (5)*(10), 10 times, (think of covering up the factor 5, and the 10 remains), and 10 times 3 is 30. And 10 goes into (5)*(10), 5 times, and 5 times 3 is 15. So we have:

10(3) + 5(3)
  (5)*(10)

 30 + 15
(5)*(10)

Let's apply to rational expressions now. What if we have x/(x + 1) + 3x/(x + 2)? To add, we first find a common denominator. A common denominator can always be achieved by multiplying the denominators together (though we only write it as a product without actually multiplying through like we did above). 

  ______?________
    (x + 1)(x + 2)

So we can say, how many times does (x + 1) go into (x + 1)(x + 2)? It goes in (x + 2) times, times the numerator x. And how many times does (x + 2) go into (x + 1)(x + 2)? It goes in (x + 1) times, times the numerator 3x. So this would look like:

x(x + 2) + 3x(x + 1)
    (x + 1)(x + 2)

Simplify the numerator.

x² + 2x + 3x² + 3x
    (x + 1)(x + 2)

    4x² + 5x
(x + 1)(x + 2)

or 

  4x² + 5x
x² + 3x + 2

Same process holds for subtraction.