What will be the amount (in Rs. )of a sum Rs. 4,800 invested at compound interest at 15 % per annum for `2(1)/(3)` years interest being compounded yearly (nearest to Rs. 1)?

Options

A. 7 , 935
B. 6348
C. 6665
D. 6575

Answer

To find the amount of a sum of Rs. 4,800 invested at compound interest at a rate of 15% per annum for \(2 \frac{1}{3}\) years, we can follow these steps:

### Step 1: Convert the time period into years
The time period given is \(2 \frac{1}{3}\) years. We can convert this into an improper fraction:
\[ 2 \frac{1}{3} = \frac{7}{3} \text{ years} \]

### Step 2: Identify the principal and rate of interest
The principal amount (P) is Rs. 4,800, and the rate of interest (R) is 15% per annum.

### Step 3: Use the compound interest formula
The formula for calculating the amount (A) in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where:
- \(P\) = principal amount
- \(R\) = rate of interest
- \(T\) = time in years

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ A = 4800 \left(1 + \frac{15}{100}\right)^{\frac{7}{3}} \]
This simplifies to:
\[ A = 4800 \left(1 + 0.15\right)^{\frac{7}{3}} = 4800 \left(1.15\right)^{\frac{7}{3}} \]

### Step 5: Calculate \(1.15^{\frac{7}{3}}\)
To calculate \(1.15^{\frac{7}{3}}\), we can first find \(1.15^7\) and then take the cube root:
\[ 1.15^7 \approx 2.5023 \]
Now, taking the cube root:
\[ (1.15^7)^{\frac{1}{3}} \approx 2.5023^{\frac{1}{3}} \approx 1.3947 \]

### Step 6: Calculate the amount
Now substituting back into the amount formula:
\[ A \approx 4800 \times 1.3947 \approx 6697.56 \]

### Step 7: Round to the nearest rupee
Rounding \(6697.56\) to the nearest rupee gives us:
\[ A \approx 6698 \]

### Final Answer
The amount after \(2 \frac{1}{3}\) years is approximately Rs. 6698.

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