A woman investe Rs. 4000 at the start of each year at 5% compound interest per annum. How much will her investment be at the end of the `2^(nd)` year?

Options

A. Rs. 8601
B. Rs. 8615
C. Rs. 8600
D. Rs. 8610

Answer

To solve the problem of how much the woman's investment will be at the end of the 2nd year, we need to calculate the compound interest on her annual investments. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Investment
The woman invests Rs. 4000 at the start of each year at a compound interest rate of 5% per annum.

### Step 2: Calculate the Amount at the End of the First Year
For the first year, the investment of Rs. 4000 will grow at 5% interest. The formula for compound interest is:

\[ A = P(1 + r)^n \]

Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = principal amount (the initial amount of money).
- \( r \) = annual interest rate (decimal).
- \( n \) = number of years the money is invested or borrowed.

For the first investment:
- \( P = 4000 \)
- \( r = 0.05 \)
- \( n = 1 \)

Calculating the amount at the end of the first year:

\[ A_1 = 4000(1 + 0.05)^1 = 4000(1.05) = 4200 \]

### Step 3: Calculate the Amount at the End of the Second Year
At the start of the second year, she invests another Rs. 4000. Now we need to calculate the total amount at the end of the second year.

1. **Amount from the first investment (after 2 years)**:
- For the first investment, we need to calculate for 2 years:
\[ A_1 = 4000(1 + 0.05)^2 = 4000(1.05)^2 = 4000(1.1025) = 4410 \]

2. **Amount from the second investment (after 1 year)**:
- For the second investment, we calculate for 1 year:
\[ A_2 = 4000(1 + 0.05)^1 = 4000(1.05) = 4200 \]

### Step 4: Total Amount at the End of the Second Year
Now, we add the amounts from both investments to find the total amount at the end of the second year:

\[ \text{Total Amount} = A_1 + A_2 = 4410 + 4200 = 8610 \]

### Final Answer
The total amount at the end of the 2nd year will be Rs. 8610.

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