Find the amount of Rs. 8000 in 1 year at 5% per annum compound interest payable half-yearly

Answer

To find the amount of Rs. 8000 in 1 year at 5% per annum compound interest payable half-yearly, we will follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 8000
- 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 is:
\[ R_{half-yearly} = \frac{5\%}{2} = 2.5\% \]
- The time for half-yearly compounding over 1 year is:
\[ T_{half-yearly} = 2 \text{ (since there are 2 half-year periods in 1 year)} \]

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

Substituting the values:
\[ A = 8000 \left(1 + \frac{2.5}{100}\right)^2 \]

### Step 4: Calculate the amount
First, simplify the expression inside the parentheses:
\[ A = 8000 \left(1 + 0.025\right)^2 = 8000 \left(1.025\right)^2 \]

Now calculate \( (1.025)^2 \):
\[ (1.025)^2 = 1.050625 \]

Now substitute this back into the amount formula:
\[ A = 8000 \times 1.050625 \]
\[ A = 8405 \]

### Step 5: Calculate the compound interest
The compound interest (CI) is given by:
\[ CI = A - P \]
Substituting the values we found:
\[ CI = 8405 - 8000 = 405 \]

### Final Answer
The amount after 1 year is Rs. 8405, and the compound interest earned is Rs. 405.

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