Veer Invested X Rs. in SBI at the rate of 18% for 2 year and obtained a total simple interest of 6750 Rs. If he invested 2250 Rs. more for same period time at the rate of `14(2)/(7)%` for first year and at `16(2)/(3)` for `2^(nd)` year on compound annually. Then find the total compound interest obtained by Veer after 2 year?

Options

A. 6000 Rs.
B. 5000 Rs.
C. 3000 Rs.
D. 7000 Rs.

Answer

To solve the problem step by step, we will break it down into parts: calculating the principal amount from the simple interest and then determining the compound interest from the second investment.

### Step 1: Calculate the Principal Amount (X) from Simple Interest
The formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest
- \(P\) = Principal amount
- \(R\) = Rate of interest
- \(T\) = Time in years

From the question:
- \(SI = 6750\) Rs
- \(R = 18\%\)
- \(T = 2\) years

Plugging in the values:
\[ 6750 = \frac{X \times 18 \times 2}{100} \]

### Step 2: Rearranging to find X
Rearranging the equation to solve for \(X\):
\[ 6750 = \frac{36X}{100} \]
\[ 6750 \times 100 = 36X \]
\[ 675000 = 36X \]
\[ X = \frac{675000}{36} = 18750 \text{ Rs} \]

### Step 3: Calculate the Total Amount Invested in the Second Scheme
Veer invested an additional 2250 Rs, so the total principal for the second investment is:
\[ Total\ Principal = 18750 + 2250 = 21000 \text{ Rs} \]

### Step 4: Calculate Compound Interest for the Second Investment
The rates for the two years are:
- First year: \(14 \frac{2}{7}\% = \frac{104}{7}\%\)
- Second year: \(16 \frac{2}{3}\% = \frac{50}{3}\%\)

#### First Year Calculation
For the first year, the amount after interest will be:
\[ A_1 = P \left(1 + \frac{R_1}{100}\right) = 21000 \left(1 + \frac{104/7}{100}\right) \]
Calculating the first year's interest:
\[ A_1 = 21000 \left(1 + \frac{104}{700}\right) = 21000 \left(\frac{700 + 104}{700}\right) = 21000 \left(\frac{804}{700}\right) \]
\[ A_1 = 21000 \times \frac{804}{700} = 21000 \times 1.14857 \approx 22019.57 \text{ Rs} \]

#### Second Year Calculation
Now for the second year:
\[ A_2 = A_1 \left(1 + \frac{R_2}{100}\right) = 22019.57 \left(1 + \frac{50/3}{100}\right) \]
Calculating the second year's interest:
\[ A_2 = 22019.57 \left(1 + \frac{50}{300}\right) = 22019.57 \left(\frac{300 + 50}{300}\right) = 22019.57 \left(\frac{350}{300}\right) \]
\[ A_2 = 22019.57 \times \frac{350}{300} \approx 22019.57 \times 1.16667 \approx 25796.67 \text{ Rs} \]

### Step 5: Calculate the Total Compound Interest
The total compound interest (CI) earned is given by:
\[ CI = A - P \]
Where \(A\) is the total amount after 2 years and \(P\) is the principal amount invested in the second scheme.
\[ CI = 25796.67 - 21000 = 4796.67 \text{ Rs} \]

### Conclusion
The total compound interest obtained by Veer after 2 years is approximately 4796.67 Rs.