A, B and C invested capitals in the ratio of 4 : 6 : 9. At the end of the business term, they received the profit in the ratio of 2 : 3 : 5. Find the ratio of their time for which they contributed their capitals.

Let B remained in the business for N months. Then,
A : B = (2250 x 12) : (2700 x N)
? 2 : 1 = (27000 : 2700N) = (10 : N)
? 10/N = 2/1 [ ? Ratio of profit is 2 : 1]
? 2N = 10
? N = 10/2 = 5
Clearly, B joined after (12 - 5) = 7 months.

Let the total profit be ? N,
Then, A's share in profit = ? 3N/5
Remaining profit = N - 3N/5 = (5N - 3N) / 5 = 2N/5
? B's share in profit = ? N/5
C's share in profit = ? N/5

According to the question,
(3N/5 - N/5) = 400
2N/5 = 400; N = (400 x 5)/2 = ? 1000

Let share of Raghu be 100. Then share of Mona = 90 and share of Sonu = 99
Sonu : Mona : Raghu = 99 : 90 : 100
? Investment of Raghu = [100/(99 + 90 + 100)] x 5780
= (100/289) x 5780 = ? 2000

Here, p₁ : p₂ : p₃ = 5 : 3 : 1 and x₁ : x₂ : x₃ = 5 : 6 : 8
According to the formula,
Required ratio = p₁/x₁ : p₂/x₂ : p₃/x₃
= 5/5 : 3/6 : 1/8
= 1 : 1/2 : 1/8 = 8 : 8/2 : 8/8 = 8 : 4 : 1

Ratio of rents of A, B, C and D
= (18 x 4) : (25 x 2) : (28 x 5) : (21 x 3)
= 72 : 50 : 140 : 63

Let the total rent be N.
Then, A's share = [72N / (72 + 50 + 140 + 63)] = 72N/325

According to the question. 72N/325 = 720
? N = (720 x 325)/72 = 3250
? Total Rent = ? 3250

Let investment of Aarti during first year = 5a
Investment of Vinita during first year = 7a
Investment of kamla during first year = 6a

Then, their investments during second year are
? (126/100) x 5a : (120/100) x 7a : (115/100) x 6a
= 630 : 840 : 690 = 21 : 28 : 23

For management, money received by Anil = 10% of 9000 = ? 900
Balance = ? (9000 - 900) = ? 8100
Now, ratio of investments =12000 : 20000 = 3 : 5

? Anil's share = ? 8100 x 3/(3 + 5)
= ? 8100 x 3/8 = ? 3037.50

? Amount received by Anil = ? 900 + 3037.50
= ? 3937.50

Let the total capital be ? R and Babloo's money is used for N months.
Capital of Aayush = R/4
? Capital of Babloo = (R - R/4) = 3R/4

Ratio of investments of Aayush and Babloo
= (R/4) x 15 : (3R/4 x b) = 15/4 : 3N/4 = 5 : N

Ratio of profits of Aayush and Babloo = (1/3) : (2/3) = 1 : 2
? 5/N = 1/2
? N = 10 months

Hence, Babloo's money was used for 10 months.

Ratio of total capital of A and B = 20000 x 12 : 35000 x 12
= 240000 : 420000

Now, C gives ? 220000 to both to make the capital equal.
? A's capital : B's capital = 240000 + 200000 : 420000 + 200000
= 440000 : 440000

If A takes ? 200000 and B takes ? 20000 from C, then both have the equal capital
? Required ratio of divided amount
= 200000 : 20000
= 20 : 2 = 10 : 1

Hence. A and B should divide the amount in the ratio of 10 : 1.

P's share : Q's share = Ratio of their investments
= 13000 : 25000 = 13 : 25

Now, let P's share = 13N
Q's share = 25N

According to the question,
13N + 25N = 76000
? N = 76000/38 = 2000
? Q's share = 25N = 25 x 2000 = ? 50000