Since they work on alternate days, let us consider a period of two days.
In the period of 2 days, work done by A and B = \(\cfrac { 1 }{ 12 } +\cfrac { 1 }{ 9 } =\cfrac { 7 }{ 36 } \) 
If we consider 5 such time periods for the fraction \(\cfrac { 7 }{ 36 } \) (we consider 5 periods because
7 goes 5 times completely in 36),
work done by A and B in 5\(\times\)2(=10)days = \(5\times \cfrac { 7 }{ 36 } =\cfrac { 35 }{ 36 } \) 
Therefore, the remaining work = \(1-\cfrac { 35 }{ 36 } =\cfrac { 1 }{ 36 } \) 
 This is done by A (why?) in \(\cfrac { 1 }{ 36 } \times 12=\cfrac { 1 }{ 3 } daysB\) 
 So, the total time taken = \(10days+\cfrac { 1 }{ 3 } days=10\cfrac { 1 }{ 3 } days\) 

P’s 1 day’s work = \(\cfrac { 1 }{ 20 } \) and Q’s 1 day’s work = \(\cfrac { 1 }{ 30 } \) 
P’s work for 5 days = \(\cfrac { 1 }{ 20 } \times 5=\cfrac { 5 }{ 20 } =\cfrac { 1 }{ 4 } \) 
Therefore, the remaining work = \(1-\cfrac { 1 }{ 4 } =\cfrac { 3 }{ 4 } \) (Total work is always 1)
This remaining work was done by both P and Q. 
Work done by P and Q in a day = \(\cfrac { 1 }{ 20 } +\cfrac { 1 }{ 30 } =\cfrac { 5 }{ 60 } =\cfrac { 1 }{ 12 } \) 
Therefore, the number of days they worked together =\(\cfrac { 3/4 }{ 1/12 } =\cfrac { 3 }{ 4 } \times \cfrac { 12 }{ 1 } =9days.\)
So, Q left after 9 day from the days the work started.

If B does the work in 3 days, then A will do it in 1 day. That is, the difference is 2 days.
Here, given that the difference between A and B in completing the work is 24 days. Therefore,
A will take \(\cfrac { 24 }{ 2 } =12\) days and B will take 3 x 12 = 36 days to complete the work separately.
Hence, the time taken by A and B together to complete the work =\(\cfrac { ab }{ a+b } days\) 
= \(\cfrac { 12\times 36 }{ 12+36 } =\cfrac { 12\times 36 }{ 48 } =9days\)

(A+ B) ’s 1 day’s work = \(\cfrac { 1 }{ 16 } \) 
A’s 1 day’s work = \(\cfrac { 1 }{ 48 } \) 
ஃ B’s 1 day’s work = \(\cfrac { 1 }{ 16 } -\cfrac { 1 }{ 48 } \) 
= \(\cfrac { 3-1 }{ 48 } =\cfrac { 2 }{ 48 } =\cfrac { 1 }{ 24 } \)
ஃB alone can complete the work in 24 days

Multiplicative Factor Method:

Step 1:
Here, less days means more students. So, it is an inverse variation.
ஃ The multiplying factor is \(\cfrac { 56 }{ 27 } \) 
Step 3:
\(\therefore x=81\times \cfrac { 56 }{ 27 } \times \cfrac { 160 }{ 448 } \) 
x = 60 students.
Formula Method:
Here,P₁ = 81,D₁ = 56 and W₁ = 448
P₂ = x, D₂ = 27 and W₂ = 160
Using the formula,\(\cfrac { { P }_{ 1 }\times { D }_{ 1 } }{ { W }_{ 1 } } =\cfrac { { P }_{ 2 }\times { D }_{ 2 } }{ { W }_{ 2 } } \) 
We have,\(\cfrac { 81\times 56 }{ 448 } =\cfrac { x\times 27 }{ 160 } \) 
\(\Rightarrow x=\cfrac { 81\times 56 }{ 448 } \times \cfrac { 160 }{ 27 } \)
x = 60 students.