A sum invested at compound interest amounts to Rs 13,824 in 2 years and to Rs 16,588.80 in 3 years at a certain rate per cent per annum, interest being compounded yearly. What will be the amount of the same sum at the end of the `4^(th)` year (nearest to a rupee)?

Options

A. Rs 20,238
B. Rs 18, 248
C. Rs 19,907
D. Rs 18,579

Answer

To find the amount of the same sum at the end of the 4th year, we can follow these steps:

### Step 1: Determine the interest earned in the 3rd year
Given:
- Amount after 2 years (A2) = Rs 13,824
- Amount after 3 years (A3) = Rs 16,588.80

The interest earned in the 3rd year can be calculated as:
\[ \text{Interest in 3rd year} = A3 - A2 \]
\[ \text{Interest in 3rd year} = 16,588.80 - 13,824 = 2,764.80 \]

### Step 2: Calculate the rate of interest
The interest earned in the 2nd year can be calculated using the amount after 1 year (A1), which we can find using the formula for compound interest.

Let the principal amount be P and the rate of interest be R%.

We know:
\[ A2 = P \left(1 + \frac{R}{100}\right)^2 \]
\[ A3 = P \left(1 + \frac{R}{100}\right)^3 \]

From the interest earned in the 3rd year, we can express it as:
\[ A3 - A2 = P \left(1 + \frac{R}{100}\right)^3 - P \left(1 + \frac{R}{100}\right)^2 \]
This simplifies to:
\[ 2,764.80 = P \left(1 + \frac{R}{100}\right)^2 \cdot \frac{R}{100} \]

### Step 3: Calculate the amount at the end of the 4th year
Now, we can find the amount at the end of the 4th year using the amount at the end of the 3rd year:
\[ A4 = A3 \left(1 + \frac{R}{100}\right) \]
Substituting the values:
\[ A4 = 16,588.80 \left(1 + \frac{20}{100}\right) \]
\[ A4 = 16,588.80 \times 1.20 \]
\[ A4 = 19,906.56 \]

### Step 4: Round to the nearest rupee
Thus, rounding to the nearest rupee:
\[ A4 \approx 19,907 \]

### Final Answer:
The amount at the end of the 4th year will be Rs 19,907.

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