Rs 1600 lent at a compound interest of 5% per annum, compounded half yearly for one year will amount to:

Options

A. Rs 1640
B. Rs 1680
C. Rs 1681
D. Rs 1764

Answer

To solve the problem of finding the amount when Rs 1600 is lent at a compound interest of 5% per annum, compounded half-yearly for one year, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs 1600
- Rate of interest (R) = 5% per annum
- Time (T) = 1 year

Since the interest is compounded half-yearly, we need to adjust the rate and time accordingly.

### Step 2: Adjust the rate and time for half-yearly compounding
- The rate for half-yearly compounding = R/2 = 5%/2 = 2.5%
- The number of compounding periods in one year = 2 (since there are two half-year periods in one year)

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount
- \( R \) = Rate of interest per period
- \( n \) = Number of compounding periods

### Step 4: Substitute the values into the formula
Substituting the values we have:
- \( P = 1600 \)
- \( R = 2.5 \) (as a percentage)
- \( n = 2 \)

So,
\[ A = 1600 \left(1 + \frac{2.5}{100}\right)^2 \]

### Step 5: Simplify the expression
First, calculate \( 1 + \frac{2.5}{100} \):
\[ 1 + \frac{2.5}{100} = 1 + 0.025 = 1.025 \]

Now substitute this back into the equation:
\[ A = 1600 \times (1.025)^2 \]

### Step 6: Calculate \( (1.025)^2 \)
Calculating \( (1.025)^2 \):
\[ (1.025)^2 = 1.050625 \]

### Step 7: Calculate the final amount
Now, substitute this value back into the equation:
\[ A = 1600 \times 1.050625 \]
\[ A = 1681 \]

### Conclusion
The amount after one year will be Rs 1681.