Alright;
ADDING FRACTIONS:
Let's say you want to add 3/8 and 9/14.
First thing you want to do is find the least common multiple of the denominators (the LCD). To find the LCD, unless you clearly recognize it, you write out the denominators in their prime factorization; 8 = 2 * 2 * 2, 14 = 2 * 7. Then, to find the LCD, you count the number of different prime factors each number has, and the LARGEST number of times each prime factor appears per number. If that made sense;
8 = 2^3
14 = 2^1 * 7^1.
Therefore, you need 3 2's and 1 7 to be factors of the LCD. Therefore, your LCD will be 2^3 * 7^1 = 56.
Now that you have your LCD, you have to manipulate your fractions so that the denominators of both are the same. So, for 3/8, you need to get 56 in the denominator (and we know it's a multiple of 8, as we just showed). Therefore, you can divide 56 by 8 to find how many times 8 goes into it. The answer is 7, so therefore, you rewrite 3/8 as (3*7)/(8*7) = 21/56.
Next, you do the same thing for 9/14. 56/14 = 4, so (9*4)/(14*4) = 36/56.
Next, you add 21/56 + 36/56. You add JUST the numerators, keeping the DENOMINATOR the same. (21 + 36) / 56 = 57/56. If you want, you can rewrite this as a mixed number; 1 and 1/56. Although it may seem quite confusing at first, it gets MUCH easier with practice.
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SUBTRACTING Fractions
Let's say we wanna do 17/60 - 5/126.
Remember the first step? We have to find the LCD.
So;
60 = 2^2 * 3 * 5
126 = 2 * 3^2 * 7.
So, we add up the HIGHEST number of each prime factor in the denominators; we need two 2's, two 3's, a 5, and a 7 to make up the LCD. So, the LCD = 4 * 9 * 5 * 7 = 1260 (yikes).
But, we must solve the problem! 1260/60 = 21. 1260/126 = 10.
So; we have (17*21)/1260 - (10*5)/1260.
357/1260 - 50/1260 = 307/1260.
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MULTIPLYING Fractions
Luckily, here is where it gets easy
Let's say you want to multiply
3/4 * 8/11 * 33/21.
Generally, you want to cancel stuff (meaning divide by n/n, yielding 1) before you do the multiplication, but it doesn't really matter. I'd like to do that first. So, this expression can be rewritten as 8/4 * 33/11 * 3/21 (you still have 3 * 8 * 33 in the numerator and 4 * 11 * 21 in the denominator). So, you can rewrite this as 2 * 3 * 1/7 (since we know that 8/4 = 2 and 33/11 = 3). Therefore, the answer is 6/7.
However, a more straightforward way that leads you straight to the answer, then requires simplification, is just multiply the numerators and multiply the denominators. Literally. It's really simple. So, you have;
(3 * 8 * 33)/(4 * 11 * 21) = 792 / 924. Next, you want to simplify this. If you don't happen to know that both are divisible by 132, you can keep dividing by simple factors that go into both, i.e; 4 (198/231), then 3 (66/77), then 11 (6/7) and you'll end up with the same answer.
Quite simple really, once you get used to it.
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Let's say you want to divide 3/7 by 2/5.
So, you have (3/7) / (2/5).
First step is to flip the division sign to a multiplication sign, and find the reciprocal of the fraction on the RIGHT (the divisor). So, you'd have:
(3/7) * (5/2).
Then, you do your multiplication (which you just reviewed), and get 15/14, or 1 1/14.
Voila!
Basically, all it comes down to is practice, practice, and more practice. Hope this helped!