A sum of Rs. 6,600 amount to Rs. 8,756 in `4(2)/(3)` years at `x%` p.a. simple interest. What will be the same amount to if invested at (x+3)% p.a. for `2(1)/(2)` years, interest compounded yearly (nearest To Rs. 1)?

Options

A. Rs. 8,066
B. Rs. 8,385
C. Rs. 8,458
D. Rs. 8,175

Answer

To solve the problem step by step, we will first determine the simple interest rate \( x \) and then calculate the compound interest for the new rate \( (x + 3)\% \).

### Step 1: Calculate the Simple Interest
We know that:
- Principal (P) = Rs. 6600
- Amount (A) = Rs. 8756
- Time (T) = \( 4 \frac{2}{3} \) years = \( \frac{14}{3} \) years

First, we find the simple interest (SI):
\[ SI = A - P = 8756 - 6600 = 2156 \]

### Step 2: Use the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R \) is the rate of interest per annum.

Substituting the known values:
\[ 2156 = \frac{6600 \times x \times \frac{14}{3}}{100} \]

### Step 3: Solve for \( x \)
Rearranging the equation to find \( x \):
\[ 2156 = \frac{6600 \times x \times 14}{300} \]
\[ 2156 \times 300 = 6600 \times x \times 14 \]
\[ 646800 = 92400x \]
\[ x = \frac{646800}{92400} = 7 \]

### Step 4: Determine the New Rate
Now, we find the new rate:
\[ x + 3 = 7 + 3 = 10\% \]

### Step 5: Calculate the Compound Interest
Now we need to find the amount if Rs. 6600 is invested at \( 10\% \) for \( 2 \frac{1}{2} \) years, compounded yearly.

Convert \( 2 \frac{1}{2} \) years to a fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \text{ years} \]

The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = 6600 \left(1 + \frac{10}{100}\right)^{\frac{5}{2}} \]
\[ A = 6600 \left(1 + 0.1\right)^{2.5} \]
\[ A = 6600 \left(1.1\right)^{2.5} \]

### Step 6: Calculate \( (1.1)^{2.5} \)
Using a calculator or logarithm tables:
\[ (1.1)^{2.5} \approx 1.27628 \]

### Step 7: Calculate the Final Amount
Now substituting back:
\[ A \approx 6600 \times 1.27628 \approx 8414.85 \]

### Step 8: Round to the Nearest Rupee
Rounding off to the nearest rupee:
\[ A \approx 8415 \]

### Final Answer
The amount will be approximately Rs. 8415.