A sum becomes Rs. 2880 in two years and Rs. 4147.2 in four years when compounded annually. If same sum is invested at 12% p.a. S.I. for 5 years, then find the amount after 5 years.

Options

A. A)3000 Rs.
B. B)3600 Rs.
C. C)2400 Rs.
D. D)3200 Rs.

Answer

To solve the problem step by step, we will first determine the principal amount using the information provided about compound interest, and then we will calculate the amount after 5 years using simple interest.

### Step 1: Determine the Principal Amount
We know that the sum becomes Rs. 2880 in 2 years and Rs. 4147.2 in 4 years when compounded annually.

1. Let the principal amount be \( P \).
2. After 2 years, the amount is given by:
\[ A_2 = P(1 + r)^2 = 2880 \]
3. After 4 years, the amount is given by:
\[ A_4 = P(1 + r)^4 = 4147.2 \]
4. We can find the ratio of the two amounts:
\[ \frac{A_4}{A_2} = \frac{P(1 + r)^4}{P(1 + r)^2} = \frac{4147.2}{2880} \]
Simplifying gives:
\[ (1 + r)^2 = \frac{4147.2}{2880} \]
5. Calculate the right side:
\[ (1 + r)^2 = 1.44 \]
6. Taking the square root:
\[ 1 + r = \sqrt{1.44} = 1.2 \]
7. Thus, the rate \( r \) is:
\[ r = 1.2 - 1 = 0.2 \quad \text{or} \quad 20\% \]
8. Now substitute \( r \) back into the equation for \( A_2 \):
\[ P(1.2)^2 = 2880 \]
\[ P \cdot 1.44 = 2880 \]
9. Solving for \( P \):
\[ P = \frac{2880}{1.44} = 2000 \]

### Step 2: Calculate Simple Interest for 5 Years
Now that we have the principal amount, we can calculate the amount after 5 years at 12% per annum simple interest.

1. The formula for simple interest is:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
Where:
- \( P = 2000 \)
- \( r = 12\% \)
- \( t = 5 \) years
2. Substituting the values:
\[ \text{SI} = \frac{2000 \times 12 \times 5}{100} = \frac{120000}{100} = 1200 \]

### Step 3: Calculate Total Amount After 5 Years
1. The total amount \( A \) after 5 years is given by:
\[ A = P + \text{SI} \]
\[ A = 2000 + 1200 = 3200 \]

### Final Answer
The amount after 5 years is Rs. 3200.

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