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Adding fractions may seem tricky at first, but once you get the hang of it, it's quite simple. The key to adding fractions is to ensure that both fractions have the same denominator. Once the denominators are equal, you only need to add the numerators together!
For example, in the problem \(\frac{1}{3}\) + \(\frac{1}{9}\), the fractions initially have different denominators (3 and 9). To add them, we must first make the denominators the same. After finding a common denominator (in this case, 9), we convert \(\frac{1}{3}\) to \(\frac{3}{9}\).

Now we can easily add the fractions since they both have the denominator 9: \(\frac{3}{9} + \frac{1}{9} = \frac{4}{9}\).

A common denominator is a shared multiple of the denominators of two or more fractions. It allows you to add or subtract fractions smoothly. To find a common denominator, look for the least common multiple (LCM) of the denominators.

In our exercise, we need to add \(\frac{1}{3}\) + \(\frac{1}{9}\). The denominators here are 3 and 9. The LCM of 3 and 9 is 9. This means we can convert both fractions so they have the same denominator.

Here's the conversion:
\(\frac{1}{3} \) becomes \(\frac{3}{9}\) because 1 times 3 is 3, and 3 times 3 is 9. \(\frac{1}{9}\) stays \(\frac{1}{9}\) since its denominator is already 9. Now, both fractions have a common denominator of 9, making it easy to add them.

Subtracting negative numbers can be confusing at first, but it follows a straightforward rule: subtracting a negative is the same as adding its positive counterpart. This is because two negatives make a positive.

For the given problem, \(\frac{1}{3} - (-\frac{1}{9})\), we can re-write it using the rule above: \(\frac{1}{3} + \frac{1}{9}\).

In other words, whenever you see a subtraction sign in front of a negative number, you can replace the subtraction sign with an addition sign and remove the negative sign from the number after it. This transformation simplifies the expression and allows you to proceed with familiar addition rules.

Practice converting these expressions, and soon it will become second nature!