Akshay deposited Rs. 'x' in a bank at compound interest of 30% p.a. compounded annually. At the end of 2 years, he divided the sum received from the bank between Sonu and Monu in the ratio of 9:7, respectively. Sonu invested his share in a scheme offering simple interest of 24% p.a., for 10 years while Monu invested his share in another scheme offering < simple interest of 10% p.a., for 28 years and difference between the interest received by Sonu and Monu is Rs. 1014. Abhilash invested Rs. 'x + 608' in a scheme offering compound interest of R% p.a. compounded annually for 2 years and received an interest which is equal to the interest earned by Sonu if he would have invested Rs. 1183 less than the amount received by him from Akshay at simple interest of (3R/5) % p.a. for 6 years.

If Sonu and Monu entered in a partnership with Rs. 6000 and Rs. 9000. After 5 months Monu left and Akshay joined with capital Rs. 6000 less than Monu. If at the end of year profit of Akshay was Rs. 49 less than the amount received by Monu from Akshay, then find the total profit at the end of year?

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

Solve a complex problem involving compound and simple interest with ratios and profit sharing in a partnership. Learn how Akshay, Sonu, and Monu invest and calculate interest. Suitable for Grades 11-12.

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