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When adding fractions, it's essential to ensure that the fractions involved have the same denominator. The denominator is the bottom number in a fraction and it tells us into how many equal parts the whole is divided. In this exercise, the fractions \(\frac{4}{9}\) and \(\frac{1}{9}\) both have the denominator 9, which means they are already in a form that allows us to add them directly. If the denominators were different, we would need to find a common denominator before proceeding. A common denominator is a shared multiple of the denominators of two or more fractions. To find it, you can list the multiples of each denominator and choose the smallest common one. This process ensures that the fractions are expressing parts of the same whole.

The numerator is the top number in a fraction and it indicates how many parts of the whole are being considered. In this exercise, for the fractions \(\frac{4}{9}\) and \(\frac{1}{9}\), the numerators are 4 and 1, respectively. Since the denominators are the same, we simply add the numerators together: \(\frac{4}{9} + \frac{1}{9} = \frac{4 + 1}{9} = \frac{5}{9}\). Remember that the denominator remains the same during this process because we're still working with the same sized parts of the whole. So, the key idea here is to always add the numerators while keeping the common denominator unchanged.

Once you've added the numerators and obtained a new fraction, the final step is to simplify it if possible. Simplifying a fraction involves reducing it to its simplest form, where the numerator and denominator have no common factors other than 1. In our example, the resulting fraction is \(\frac{5}{9}\). To check if it can be simplified, determine if the numerator (5) and the denominator (9) share any common factors. Here, 5 and 9 have no common factors other than 1. Hence, the fraction \(\frac{5}{9}\) is already in its simplest form. Simplifying fractions helps to keep our answers neat and makes them easier to understand and compare.