Question 21.

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:

A
B
C
D
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Text Explanation
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 $=11t=99$
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