The ratio of the number of students in the morning shift and afternoon shift of a school was 13 : 9. After 21 students moved from the morning shift to the afternoon shift, this ratio became 19 : 14. Next, some new students joined the morning and afternoon shifts in the ratio 3 : 8 and then the ratio of the number of students in the morning shift and the afternoon shift became 5 : 4. The number of new students who joined is

The correct answer is 99.

Let's solve the problem step-by-step.

Step 1: Finding the initial number of students

Let the initial number of students in the morning shift be $13x$ and the number of students in the afternoon shift be $9x$.

We are given that $21$ students moved from the morning shift to the afternoon shift. So, the new number of students in each shift becomes:

Morning shift: $13x-21$
Afternoon shift: $9x+21$

The new ratio is given as $19:14$. We can set up the following equation:

$\frac{13x-21}{9x+21}=\frac{19}{14}$

Cross-multiplying to solve for $x$:

$14(13x-21)=19(9x+21)$

$182x-294=171x+399$

$182x-171x=399+294$

$11x=693$

$x=63$

Now, let's find the number of students in each shift after the $21$ students moved:

Morning shift: $13\left(63\right)-21=819-21=798$

Afternoon shift: $9\left(63\right)+21=567+21=588$

Step 2: Calculating the number of new students

Next, some new students joined in the ratio $3:8$. Let the number of new morning students be $3y$ and new afternoon students be $8y$.

The total number of new students who joined is $3y+8y=11y$.

After these new students join, the ratio of morning to afternoon students becomes $5:4$. We can set up another equation using the previous totals:

$\frac{798+3y}{588+8y}=\frac{5}{4}$

Cross-multiplying to solve for $y$:

$4(798+3y)=5(588+8y)$

$3192+12y=2940+40y$

$3192-2940=40y-12y$

$252=28y$

$y=\frac{252}{28}=9$

Finally, we calculate the total number of new students who joined by substituting $y$ into our expression for the total new students:

$\text{Total new students}=11y=11\left(9\right)=99$

Therefore, 99 new students joined.