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

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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.