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When adding fractions, you need to make sure they have the same denominator. This common denominator is called the Least Common Denominator (LCD). The LCD is the smallest number that both original denominators can divide into without leaving a remainder. For our exercise, the denominators are 3 and 10. To find the LCD, you can list the multiples of both numbers and find the smallest one they have in common. For 3, the multiples are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so forth. For 10, the multiples are 10, 20, 30, 40, and so on. The smallest multiple they both share is 30, so our LCD is 30.

After finding the LCD, we need to convert the original fractions to equivalent fractions with this common denominator. Equivalent fractions represent the same value, even though they have different numerators and denominators. To convert a fraction to an equivalent fraction with a specific denominator, multiply both the numerator and the denominator so that the denominator becomes the LCD. For instance, to convert \( \frac{1}{3} \) to a fraction with a denominator of 30, we multiply both top and bottom by 10: \[ \frac{1}{3} \times \frac{10}{10} = \frac{10}{30} \]. Similarly, to convert \( \frac{3}{10} \) to a denominator of 30, we multiply by 3: \[ \frac{3}{10} \times \frac{3}{3} = \frac{9}{30} \]. Now both fractions have the same denominator and can be added easily.

Simplifying a fraction means writing it in its simplest form, where the numerator and denominator are as small as possible but still represent the same value. After adding the fractions, we got \( \frac{19}{30} \). To simplify this fraction, we would normally check if 19 and 30 have any common factors other than 1. In this case, they don't. Therefore, \( \frac{19}{30} \) is already in its simplest form. Always check for common factors between the numerator and the denominator to ensure your answer is fully simplified.