Socratic has warned me that this is a pretty long answer and I should consider making it more concise, but I've really covered the basics and there are four separate questions being asked!
Addition and Subtraction of fractions:
To add and subtract fractions, the numbers at the bottom of the fraction (the denominator) must be the same.
If you start with fractions with the same denominator you just add (or subtract) the numerator's (top numbers) together and place over the denominator.
If, however, the denominators are not the same, the first step is to make them the same. To do this, you need to find the LCM (Lowest Common Multiple) of the denominators numbers.
In this case, let's use this example:
Here, the two denominators are 3 and 5. The LCM of these two numbers is gained by multiplying them together to get 15.
We now know that the sum will look like this:
To determine what the numerators will be, we have to think about how we changed the denominators. To change the
Following the same process with
As with the first example, the numerators are then added together:
Multiplication of fractions:
To multiply a fraction, simply multiply the numerators and write this number at the top, and multiply the denominators and write this number at the bottom.
After the above calculation has been completed, simplification has taken place. It is, however, possible to simplify prior to multiplying which can reduce the amount of work required.
Here, if we look back at 'diagonal numbers' in the original question:
This now gives:
You can see that the same answer has been achieved but with less effort.
Division of fractions:
Division is carried out almost exactly the same as multiplication but first you 'flip' the fraction after the division sign, and then multiply it.
Yet another example!:
As before, once you've turned your division into a multiplication, reduce if possible and then carry out the multiplication on both the numerator and denominator: