Mrs. Smith's classroom can hold at the most 30 students. Her first period class had 4 boys for every 3 girls. When 3 boys and 1 girl were added to the class, the new ratio became 3 boys to 2 girls. How many total students are now in the class? Show all work and explain the process.

original class:

number of boys = 4x
number of girls = 3x

new class:
boys --- 4x+3
girls --- 3x + 1

(4x+3)/(3x+1) = 3/2
9x + 3 = 8x + 6
x = 3

original class
boys = 12
girls = 9 , notice ratio = 12:9 = 4:3

new class
boys = 15
girls = 10
, new ratio = 15:10 = 3:2

To solve this problem, we can use a system of equations.

Let's assume the number of boys in the class is 'b' and the number of girls is 'g' before the additional students are added.

According to the given information, the ratio of boys to girls in the class before the additional students are added is 4:3. This can be written as:

b/g = 4/3

We also know that the total number of students in the class cannot exceed 30. So, we can write another equation based on this limitation:

b + g ≤ 30

After 3 boys and 1 girl are added to the class, the new ratio becomes 3:2. We can write this as:

(b + 3)/(g + 1) = 3/2

Now, we can solve this system of equations to find the values of 'b' and 'g'.

First, let's solve for 'b' in terms of 'g' using the first equation:

b = (4/3)g

Now, substitute this expression for 'b' in the second equation:

(4/3)g + g ≤ 30
(7/3)g ≤ 30

Next, solve for 'g' by multiplying both sides of the inequality by the reciprocal of (7/3):

g ≤ (30 * 3)/7
g ≤ 90/7

Since 'g' represents the number of students, it must be a whole number. Therefore, we can conclude that the maximum number of girls in the class is 12.

Now, substitute this value for 'g' in the first equation to find the value of 'b':

b = (4/3) * 12
b = 16

So, before the additional students were added, there were 16 boys and 12 girls in the class.

After adding 3 boys and 1 girl, the new ratio becomes 3:2, so the new number of boys and girls can be calculated as:

New number of boys = 16 + 3 = 19
New number of girls = 12 + 1 = 13

Now, we can determine the total number of students in the class by adding the number of boys and girls:

Total number of students = 19 + 13 = 32

Therefore, there are now a total of 32 students in Mrs. Smith's classroom.