To add the fractions \( \frac{1}{7} \) and \( \frac{3}{14} \), we first need to find the least common denominator (LCD) of the two fractions.
Step 1: Find the LCD. The denominators are 7 and 14. The least common denominator of 7 and 14 is 14.
Step 2: Convert \( \frac{1}{7} \) to an equivalent fraction with a denominator of 14. To convert \( \frac{1}{7} \) to a fraction with a denominator of 14, we can multiply the numerator and denominator by 2:
\[ \frac{1}{7} = \frac{1 \times 2}{7 \times 2} = \frac{2}{14} \]
Step 3: Now we have: \[ \frac{1}{7} = \frac{2}{14} \] \[ \frac{3}{14} = \frac{3}{14} \text{ (already in the correct form)} \]
Step 4: Add the two like fractions. Now that both fractions have the same denominator, we can add them:
\[ \frac{2}{14} + \frac{3}{14} = \frac{2 + 3}{14} = \frac{5}{14} \]
Final Answer: The sum \( \frac{1}{7} + \frac{3}{14} = \frac{5}{14} \).
The fraction \( \frac{5}{14} \) is already in simplest form, so that is the final answer.