In class 6T, when only one girl stands up and the rest of the children are sitting down, the number of boys sitting down is $\frac{1}{2}$ the number of girls sitting down. When only one boy stands up and the rest of the children are sitting down, the ratio of the number of girls sitting down to the number of boys sitting down is 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 = $9 + 9 = 18$.