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

121

110

✅ Correct Option: 3

Let the common multiplier be $x$ .

Morning shift $= 13x$

Afternoon shift $= 9x$

After 21 students move from morning to afternoon, the new ratio becomes $19 : 14$ .

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

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

$182x - 294 = 171x + 399$

$11x = 693$

$x = 63$

After the transfer:

Morning $= 13(63) - 21 = 798$

Afternoon $= 9(63) + 21 = 588$

New students join morning and afternoon in the ratio $3 : 8$ . This means for every 3 students joining morning, 8 join afternoon. Let the new students joining be $3y$ and $8y$ respectively.

After joining, the ratio becomes $5 : 4$ .

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

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

$3192 + 12y = 2940 + 40y$

$252 = 28y$

$y = 9$

Total new students $= 3y + 8y$

$= 11y$

$= 11(9)$

$= 99$

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