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

Let M=13k, A=9k. 

After 21 students were moved $$\frac{13k-21}{9k+21}=\frac{19}{14}\Rightarrow 14(13k-21)=19(9k+21)$$

So $$182k-294=171k+399\Rightarrow 11k=693\Rightarrow k=63$$

So, the number of students in the morning and afternoon shifts are $$819-21=798, 567+21=588$$ respectively.

Let's assume that $$3t$$ and $$8t$$ students joined the respective sessions. 

$$\dfrac{798+3t}{588+8t}=\dfrac{5}{4}\Rightarrow 4(798+3t)=5(588+8t)$$

So $$3192+12t=2940+40t\Rightarrow 28t=252\Rightarrow t=9$$

Number of new students $$=11t=99$$