Since they all work for the same number of days, the ratio in which they share the money is equal to the ratio of their work done per day.
That is,\(\cfrac { 1 }{ 4 } :\cfrac { 1 }{ 6 } :\cfrac { 1 }{ 10 } =\cfrac { 15 }{ 60 } :\cfrac { 10 }{ 60 } :\cfrac { 6 }{ 60 } =15:10:6\) 
Here, the total parts = 15 + 10 + 6 = 31
Hence, A’ s share =\(\cfrac { 15 }{ 31 } \times 3100=Rs.1500\), B; share = \(\cfrac { 10 }{ 31 } \times 3100=Rs.1000\) and
C’ s share is Rs.3100 - (Rs.1500 + Rs.1000) = Rs.600

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 = \(10days+\cfrac { 1 }{ 3 } days=10\cfrac { 1 }{ 3 } days\) 

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

Proportion Method:

Here, the unknown is the payment (x). It is to be compared with the workers and the days.
Step 1:
Here, less days means less payment. So, it is a direct proportion.
ஃ The proportion is 20 : 12 :: 6 : x   ⟶1
Step 2:
Also, less workers means less payment. So, it is a direct proportion again.
ஃ The proportion is 15 : 5 :: 6 : x  ⟶2
Step 3:
Combining 1 and 2
\(\begin{matrix} 20:12 \\ 15:5 \end{matrix}\} ::6:x\)
We know that the product of the extremes = the product of the means
So,20 x 15 = 12 x 6 x 5 ⇒ \(x=\cfrac { 12\times 6\times 5 }{ 20\times 15 } =Rs.1.2lakh\) 
Multiplicative Factor Method

Here, the unknown is the payment (x). It is to be compared with the workers and the days.
Step 1:
Here, less days means less payment. So, it is a direct proportion.
ஃ The multiplying factor is \(\cfrac { 12 }{ 20 } \) (take the reciprocal).
Step 2:
Also, less workers means less payment. So, it is a direct proportion again.
ஃ The multiplying factor is \(\cfrac { 5 }{ 15 } \) (take the reciprocal)
Step 3:
\(\therefore x=6\times \cfrac { 12 }{ 20 } \times \cfrac { 5 }{ 15 } \)
x = Rs.1.2 lakh
Formula Method
Here, P₁= 15D, D₁= 20 and W₁= 6
P₂ = 5, D₂ = 12 and W₂ = x
Using the formula,\(\cfrac { { P }_{ 1 }\times { D }_{ 1 } }{ { W }_{ 1 } } =\cfrac { { P }_{ 2 }\times { D }_{ 2 } }{ { W }_{ 2 } } \) 
We have,\(\cfrac { 15\times 20 }{ 6 } =\cfrac { 5\times 12 }{ x } \) 
\(\Rightarrow x=\cfrac { 5\times 12\times 6 }{ 15\times 20 } =Rs.12lakh\)