The problem you've given is complex and multi-step, involving multiple concepts from compound interest, simple interest, and profit-sharing partnerships. To solve this problem, we will break it down into smaller parts, solve each piece, and then combine the results.

Step 1: Calculate the total amount after 2 years of compound interest

Akshay deposited Rs. $x$ at 30% p.a. compounded annually for 2 years. The formula for compound interest is:

$A = P \left( 1 + \frac{r}{100} \right)^t$

Where:

• $A$ = Amount after interest
• $P$ = Principal (initial deposit, which is $x$)
• $r$ = Interest rate per annum (30%)
• $t$ = Time (2 years)

$A = x \left( 1 + \frac{30}{100} \right)^2 = x \left( 1.3 \right)^2 = x \times 1.69$

So, after 2 years, the amount received by Akshay is $1.69x$.

Step 2: Divide the amount between Sonu and Monu

The total amount $1.69x$ is divided between Sonu and Monu in the ratio 9:7. Let Sonu’s share be $S$ and Monu’s share be $M$. We know:

$S = \frac{9}{9+7} \times 1.69x = \frac{9}{16} \times 1.69x = 0.950625x$ $M = \frac{7}{16} \times 1.69x = 0.739375x$

Step 3: Calculate the interest received by Sonu and Monu

• Sonu: Sonu invested his share $0.950625x$ at simple interest of 24% p.a. for 10 years. The formula for simple interest is:

$I = P \times \frac{r \times t}{100}$ $I_{\text{Sonu}} = 0.950625x \times \frac{24 \times 10}{100} = 0.950625x \times 2.4 = 2.2815x$

• Monu: Monu invested his share $0.739375x$ at simple interest of 10% p.a. for 28 years.

$I_{\text{Monu}} = 0.739375x \times \frac{10 \times 28}{100} = 0.739375x \times 2.8 = 2.07025x$

Step 4: Find the difference between the interest received by Sonu and Monu

The difference in interest between Sonu and Monu is Rs. 1014, so:

$I_{\text{Sonu}} - I_{\text{Monu}} = 1014$ $2.2815x - 2.07025x = 1014$ $0.21125x = 1014$ $x = \frac{1014}{0.21125} = 4800$

So, Akshay initially deposited Rs. 4800.

Step 5: Solve for the interest earned by Abhilash

Abhilash invested Rs. $x + 608 = 4800 + 608 = 5408$ at compound interest of $R \%$ p.a. for 2 years. The interest earned by Abhilash is equal to the interest Sonu would have earned if he invested $1183$ less than his amount at simple interest of $\frac{3R}{5} \%$ for 6 years.

Let’s first calculate the amount Sonu would have invested:

$\text{Sonu's investment} = 0.950625x - 1183 = 0.950625 \times 4800 - 1183 = 4563 - 1183 = 3380$

Now, the simple interest formula is:

$I = P \times \frac{r \times t}{100}$ Substituting $P = 3380$, $r = \frac{3R}{5}$, and $t = 6$:

$I_{\text{Sonu}} = 3380 \times \frac{\frac{3R}{5} \times 6}{100} = 3380 \times \frac{18R}{500} = 121.68R$

Step 6: Partnership details

Sonu and Monu entered into a partnership with Rs. 6000