please answer the following?

Let n be the original number of Mrs. Hoffman's fifth-grade students.
Original number of boys = (2/3)n
Original number of girls = [1 - (2/3)]n = (1/3)n

(2/3)n - 6 = (1/3)n + 6
(2/3)n - (1/3)n = 12
(1/3)n = 12
n = 36

Originally, there are 36 students in Mrs. Hoffman's class.  Among them, there are 24 boys and 12 girls.
After 6 boys going to the other class and 6 girls come from that class into Ms. Hoffman's class, there are 18 boys and 18 girls.

Two-thirds of Mrs. Hoffman’s fifth-grade students are boys.
To make the number of boys and girls equal,
6 boys go to the other fifth-grade class,
and 6 girls come from that class into Mrs. Hoffman’s class.
Now one-half of her students are boys.
How many students are in Mrs. Hoffman’s class?
B : G = 2 : 1
(B - 6) : (G + 6) = 1 : 1
Solutions:
B = 24, G = 12
There are 36 students in Mrs. Hoffman's class.

Let b = number of boys originally
Let g = number of girls originally

Then the total number of kids in the class is (b + g)

If 2/3 of the class are boys, then we can say:

b = (2/3)(b + g)

Let's simplify this.  Starting with multiply both sides by 3:

3b = 2(b + g)
3b = 2b + 2g
b = 2g

Now if 6 boys are removed and 6 girls are added, then we have:

number of boys = b - 6
number of girls = g + 6

The total number of students is still (b + g) but the boys are now 1/2 of the class, so set up another equation as we did above using the new expression for the number of boys:

b - 6 = (1/2)(b + g)

Multiply both sides by 2:

2b - 12 = b + g
b = g + 12

We now have a system of two equations and two unknowns that we can solve for.

Both expressions are equal to "b" so both expressions are equal to each other:

b = 2g and b = g + 12
2g = g + 12
g = 12

Now we can solve for b:

b = g + 12
b = 12 + 12
b = 24

Before moving students around, there were 24 boys and 12 girls in the class.