​The ratio of the number of students in the morning shift and the afternoon shift of a school was 13 : 9. After 'n' students moved from the morning shift to the afternoon shift, the said ratio became 19 : 14. Now 153 new students sought admission in the school, and these students joined the morning and the afternoon shifts in the ratio 6 : 11 respectively. If the ratio of the number of students in the morning shift and the afternoon shift of the school after the new admissions became 24 : 19, find the value of 'n'.​

Correct option is D

Given:

Initial ratio of students:

Morning shift (M) : Afternoon shift (A) = 13 : 9

After moving 'n' students from M to A, the new ratio becomes 19 : 14

153 new students join in the ratio 6 : 11 (Morning : Afternoon).

Final ratio after new admissions: 24 : 19

Solution:

Let initial students be M = 13x , A = 9x

After transfer:

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

182x - 14n = 171x + 19n

​$$11x = 33n \implies x = 3n$$​​

New admissions:

Morning: 54 , Afternoon: 99

Final ratio:

​$$\frac{13(3n) - n + 54}{9(3n) + n + 99} = \frac{24}{19}$$​​

​$$\frac{38n + 54}{28n + 99} = \frac{24}{19}$$​​

​$$19(38n + 54) = 24(28n + 99)$$​​

​$$50n = 1350 \implies n = 27$$

Thus the correct option is (d) 27.