To find the sum of the fractions \( \frac{1}{18} + \frac{1}{68} \), we need to find a common denominator.
The least common multiple (LCM) of 18 and 68 can be calculated as follows:
- The prime factorization of 18 is \( 2 \times 3^2 \).
- The prime factorization of 68 is \( 2^2 \times 17 \).
The LCM is calculated by taking the highest power of each prime factor involved:
- For 2: the highest power is \( 2^2 \).
- For 3: the highest power is \( 3^2 \).
- For 17: the highest power is \( 17^1 \).
Thus, \( \text{LCM}(18, 68) = 2^2 \times 3^2 \times 17 = 4 \times 9 \times 17 = 612 \).
Now, we can rewrite each fraction with the common denominator of 612:
\[ \frac{1}{18} = \frac{34}{612} \quad \text{(since \( 18 \times 34 = 612 \))} \] \[ \frac{1}{68} = \frac{9}{612} \quad \text{(since \( 68 \times 9 = 612 \))} \]
Now we can add the fractions:
\[ \frac{34}{612} + \frac{9}{612} = \frac{34 + 9}{612} = \frac{43}{612} \]
Thus, the sum of the fractions is:
\[ \boxed{\frac{43}{612}} \]