Understanding Addition of Fractions:

Let's start by looking at the addition of fractions using number lines.

When and are plotted on number lines, the subdivisions (sections) on the number lines are different. One of the subdivision is "thirds" and the other is "fourths". Because of this difference in the size of these sections, we have no way of combining (or adding) them together. It would be like adding apples and dogs and wondering what you actually had when you were done.

There must be a way to "fix" the subdivisions (sections) so that they could be the same size. The trick is to make more equal subdivisions until we have the same equal number on each line.

On the "thirds" line, if we cut each "third" into four equal sections we will get 12 in total.
On the "fourths" line, if we cut each "fourth" into three equal sections we will get 12 in total.
(This number 12 will be called our common denominator.)

Now that both lines contain subdivisions of the same size (both have 12 sections), we can express our fractions using this new subdivision size of 1/12. Then we can add the fractions together (since they are representatives of the same size sections). Now we are adding apples to apples.

Let's combine our number lines onto one line to demonstrate this sum.

Notice that the, when multiplied top and bottom by 4, creates our new fraction .
(4 was the number of subdivisions we created for each "third" section)

Notice that the , when multiplied top and bottom by 3, creates our new fraction .
(3 was the number of subdivisions we created for each "fourth" section)

 Same Denominators:
When the fractions have the same denominators, addition and subtraction are easy.
Think of it as "already dealing with equal sections" on the number line.

When adding (or subtracting) fractions with the same denominators,
just add (or subtract) the numerators.

Since the denominator are the same, the answer can be found by adding the numerators (tops).
 This answer is in simplest form.
There are no numbers that divide exactly into both 5 and 7.
(5 and 7 are prime numbers)

These fractions have the same denominators, so add the numerators.
 This answer can be simplified further. This is the answer in simplest form.

When dealing with mixed numbers,

Method 1: Separate parts horizontally

Method 2: Line up vertically

 Whether you use method 1 or method 2 to solve this problem, the final answer can be reduced (simplified) to

 Different Denominators:
When fractions have different denominators, you must find a common denominator. Think of it as "finding the needed number of subdivisions" for the number line.

When adding (or subtracting) fractions with different denominators,
you must
find a common denominator before adding (or subtracting).

 The least common denominator for 4 and 3 is 12.

If an answer is an improper fraction, consider rewriting it as a mixed number.

 The least common denominator for 8 and 5 is 40.

Adding fractions depends upon whether a common denominator is present.
You need that equal division of space (equal sections) discussed on this page.
The Fundamental Law of Fractions (applied to adding fractions) discusses this
need of a common denominator.

 Fundamental Law of Fractions (applied to adding fractions): (Note: bd will not necessarily be the "least" common denominator)

In plain English, this law is telling us that we can find a common denominator by finding the product (the multiplication) of the two existing denominators. The numerators must then be adjusted accordingly.
(bd is the product of the two denominators)