Let the total profit be Rs. 100.

After paying to charity, A's share = Rs.  \(\left(95\times\frac{3}{5}\right)\) = Rs.57.

If A's share is Rs. 57, total profit = Rs. 100.

If A's share Rs. 855, total profit =    \(\left(\frac{100}{57}\times855\right)\)  = 1500.

For managing, A received = 5% of Rs. 7400 = Rs. 370.

Balance = Rs. (7400 - 370) = Rs. 7030.

Ratio of their investments = (6500 x 6) : (8400 x 5) : (10000 x 3)

   = 39000 : 42000 : 30000

   = 13 : 14 : 10

Therefore B's share = Rs.    \(\left(7030\times\frac{14}{37}\right)\)  = RS.2660.

Ratio of initial investments =  \(\left(\frac{7}{2}:\frac{4}{3}:\frac{6}{5}\right)\)  = 105:40:36,

Let the initial investments be 105x, 40x and 36x.

Therefore A : B : C = \(\left(105x\times4+\frac{150}{100}\times105x\times8\right)\)      : (40x x 12) : (36x x 12)

= 1680x : 480x : 432x = 35 : 10 : 9.

Hence, B's share = Rs.      \(\left(21600\times\frac{10}{54}\right)\)  = Rs. 4000.

Let C = x.

Then, B = x + 5000 and A = x + 5000 + 4000 = x + 9000.

So, x + x + 5000 + x + 9000 = 50000

 3x = 36000

 x = 12000

A : B : C = 21000 : 17000 : 12000 = 21 : 17 : 12.

Therefore A's share = Rs.  \(\left(35000\times\frac{21}{50}\right)\)   = Rs.14,700.

Let their investments be Rs. x for 14 months, Rs. y for 8 months and Rs. z for 7 months respectively.

Then, 14x : 8y : 7z = 5 : 7 : 8.

Now, \(\frac{14x}{8y} = \frac{5}{7} \Leftrightarrow 98x=40y \Leftrightarrow y = \frac{49}{20}x \)

And , \(\frac{14x}{7z} = 112x = 35z \Leftrightarrow z = \frac{112}{35} =\frac{16}{5}x.\)

Therefore x : y : z = x :  \(\frac{49}{20}x:\frac{16}{5}x\) = 20 : 49 : 64.

Let B's capital be Rs. x.

Then, \(\left(\frac{3500\times12}{7x} = \frac{2}{3}\right)\)

 14x = 126000

 x = 9000.

A : B = \(\left[4x\times3+\left(4x-\frac{1}{4}\times4x\right)\times7\right] : \left[5x\times3+\left(5x-\frac{1}{5}\times5x\right)\times7\right] \)

= (12x + 21x) : (15x + 28x)

   = 33x :43x

   = 33 : 43.

Therefore A's share = Rs. \(\left(760\times\frac{33}{36}\right)\)  =  Rs. 330.

Let the initial investments of A and B be 3x and 5x.

A : B : C = (3x x 12) : (5x x 12) : (5x x 6) = 36 : 60 : 30 = 6 : 10 : 5.

A : B : C = (10 x 7) : (12 x 5) : (15 x 3) = 70 : 60 : 45 = 14 : 12 : 9.

Therefore C's rent = Rs.   \(\left(175\times\frac{9}{35}\right)\) = Rs. 45.

A : B : C = (20,000 x 24) : (15,000 x 24) : (20,000 x 18) = 4 : 3 : 3.

Therefore B's share = Rs. \(\left(25000\times\frac{3}{10}\right)\)  = Rs. 7,500.