A starts a business with ? 4000. B joins him after 3 months with ? 8000. C puts a sum of ? 20000 in the business for 2 months only. At the end of year, the business gives a profit of ? 16800. How should the profit be divided amongst them?

Total profit - Remuneration = Balance profit This balance profit is divided in proportion to their investment
? (Balance Profit of A) / (Balance Profit of B) = (Investment of A) / (investment of B)
? (390 - 10 x 12) / balance of Profit of B = 3000/4000 = 3/4
(Since remuneration of A is Rs. 10 per month)
? Balance profit of B = (4 x 270)/3 = Rs. 360
Since B does not get any remuneration, hence B receives Rs. 360 at the end of the year.

Ratio of rent = 6 : 10 : 12 = 3 : 5 : 6
? Pandeyji has to pay = Rs. 350 x (6/14) = Rs. 150

For management A receive = Rs. 960
Balance Amount = Rs. (9600 - 960) = Rs. 86400
Ratio of their investment = 12000 : 20000 = 3 : 5
? A's share = Rs. 8640 x (3/8) = Rs. 3240
So, A receive = Rs. (3240 + 960) = Rs. 4200

Sanjay's investment for 1 months = 15000 x 12 = Rs. 180000
Komal's investment for 1 months = 15000 x 8 + (15000 - 10000) x 4
= 120000 + 20000
= Rs. 140000

? Ratio between the investment of Sanjay and Komal
= 180000 : 140000 = 9 : 7
? Sanjay's profit = [9/(9 + 7)] x 32000 = Rs. 18000

Let the total profit be Rs. 100
After paying to Charity,
A's share = Rs. 95 x (3/5) = Rs. 57
If A's share is Rs. 57, then total profit = Rs. 100
If A's share is Rs. 855, then total profit = Rs. 855 x (100/51) = Rs. 1500

Let R's capital = 1
Then, Q's capital = 4
2 (P's capital) = 3 (Q's capital) = 3 x 4 = 12
? P's capital = 12/2 = 6
? P's share : Q's share : R's share = 6 : 4 : 1
Thus, Q's share profit = { 4/(6 + 4 + 1)} x 148500
= (4/11) x 148500
= 4 x 13500
= ? 54000

Let investment of P = ? K
Then, investment of N of = ? (K + 5000)
and investment of M = (K + 5000) + 4000 = ? (K + 9000)

According to the question.
K + (K + 5000) + (K + 9000) = 50000
? 3K + 14000 = 50000
? 3K = 50000 - 14000 = 36000
? K = 36000/3 = 12000

Clearly, investment of P = ? 12000
Investment of N = (K + 5000) = 12000 + 5000 = ? 17000
Investment of M = (K + 9000) = 12000 + 9000 = ? 21000
M's share : N's share : P's share = 21000 : 17000 : 12000
= 21 : 17 : 12

Hence, M's share = [21/(21 + 17 + 12 )] x 70000
= (21/50) x 70000 = ? 29400

Ratio of rents to be paid by P, Q and R Ratio of monthly equivalent
= (10 x 20) : (30 x 8) : (16 x 9)
= 200 : 240 : 144 = 25 : 30 : 18

Hence R's share
= [18/(25 + 30 + 18)] x 2920
= (18/73) x 2920
= 18 x 40
= ? 720

Let investment of A = N,
Investment of B = 2N,
Investment of C = 3N,

? A's share : B's share : C's share
= (N x 12) : (2N x 6) : (3N x 4)
= 12N : 12N : 12N
= 1 : 1 : 1

? C's share = [1/(1 + 1 + 1)] x 54000
= (1/3) x 54000 = ? 18000

Given, Investment of B = 1/6 of total capital
? Investments of A and C each = 1/2(1 - 1/6) of total capital
= (1/2) x (5/6) of total capital
= (5/12) of total capital

Now. A's share : B's share : C's share = (5/12) : (1/6) : (5/12) = 5 : 2 : 5
Let A's share = 5N
B's share = 2N
C's share = 5N

According to the question.
5N + 2N + 5N = 33600
? 12N = 33600
? N = 33600/12 = 2800
? Difference in the profits of B and C
= 5N - 2N = 3N = 3 x 2800 = ? 8400