在6T班，当只有一名女生站着、其余学生都坐着时，坐着的男生人数是坐着女生人数的 $\frac{1}{2}$。当只有一名男生站着、其余学生都坐着时，坐着女生人数与坐着男生人数的比是 9 : 4。

1. Formulate equations based on the two scenarios given

Let the total number of boys be $B$ and the total number of girls be $G$.
When one girl stands up, the number of girls sitting is $G - 1$ and the number of boys sitting is $B$.
$B = \frac{1}{2}(G - 1) \implies G - 1 = 2B \implies G = 2B + 1$

When one boy stands up, the number of boys sitting is $B - 1$ and the number of girls sitting is $G$.
$\frac{G}{B - 1} = \frac{9}{4} \implies 4G = 9(B - 1)$

2. Solve the simultaneous equations to find the number of boys and girls

Substitute $G = 2B + 1$ into the second equation:
$4(2B + 1) = 9(B - 1)$
$8B + 4 = 9B - 9$
$9B - 8B = 4 + 9$
$B = 13$

Substitute $B = 13$ back to find $G$:
$G = 2(13) + 1 = 27$

3. Calculate the total number of children

Total number of children = $B + G = 13 + 27 = 40$

1. Determine the initial number of boys and girls from part (a)

From part (a), let the number of boys be $B$ and girls be $G$.
When 1 girl stands: $B = \frac{1}{2}(G - 1) \implies 2B = G - 1 \implies G = 2B + 1$.
When 1 boy stands: $\frac{G}{B - 1} = \frac{9}{4} \implies 4G = 9(B - 1) = 9B - 9$.
Substitute $G$: $4(2B + 1) = 9B - 9 \implies 8B + 4 = 9B - 9 \implies B = 13$.
$G = 2(13) + 1 = 27$.
Initial boys = 13, Initial girls = 27.

2. Use the constant difference concept since an equal number of boys and girls left

Initial difference between girls and boys = $27 - 13 = 14$.
Since an equal number of girls and boys left, the difference remains constant.
New ratio of girls to boys = $9 : 2$.
Difference in ratio units = $9 - 2 = 7$ units.
So, $7 \text{ units} = 14 \implies 1 \text{ unit} = 2$.

3. Calculate the number of children who left

New number of girls = $9 \times 2 = 18$.
Number of girls who left = $27 - 18 = 9$.
Number of boys who left = $13 - (2 \times 2) = 13 - 4 = 9$.
Total number of children who left = $9 + 9 = 18$.