One-half of the students are girls, so the number of girls is $\frac{1}{2}S$.
One-seventh of the girls are in 5th grade. So the number of 5th-grade girls is $\frac{1}{7}$ of $\frac{1}{2}S$.
The fraction of 5th-grade girls out of the total number of students is:
$$ \frac{1}{7} \times \frac{1}{2} S = \frac{1}{14} S $$
The fraction of 5th-grade girls out of the total number of students is $\frac{1}{14}$.
To find the fraction of 5th-grade girls out of the total number of girls, we have:
Number of 5th grade girls = $\frac{1}{7} \times \frac{1}{2}S = \frac{1}{14}S$
Number of girls = $\frac{1}{2}S$
Fraction of 5th-grade girls out of the total number of girls is:
$$ \frac{\frac{1}{14}S}{\frac{1}{2}S} = \frac{1}{14} \times \frac{2}{1} = \frac{2}{14} = \frac{1}{7} $$
Thus, $\frac{1}{7}$ of the girls are in 5th grade.
The fraction of 5th-grade girls is $\frac{1}{7}$ of the girls, which is $\frac{1}{7}$ of $\frac{1}{2}$ of the total number of students.
Therefore, the fraction of 5th-grade girls is:
$$ \frac{1}{7} \times \frac{1}{2} = \frac{1}{14} $$
The fraction of 5th-grade girls in the school is $\frac{1}{14}$.
Final Answer: The final answer is $\boxed{1/14}$