Find the amount and the compound interest on Rs 31250 for `1(1)/(2)` years at 8% per annum, compounded half-yearly.

Options

A. Amount = Rs 35152, CI = Rs 9902
B. Amount = Rs 45152, CI = Rs 3902
C. Amount = Rs 35152, CI = Rs 3902
D. Amount = Rs 36152, CI = Rs 3902

Answer

To find the amount and the compound interest on Rs 31,250 for 1.5 years at an interest rate of 8% per annum, compounded half-yearly, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs 31,250
- Rate of interest (R) = 8% per annum
- Time (T) = 1.5 years
- Compounding frequency = half-yearly (2 times a year)

### Step 2: Convert the time into the number of compounding periods
Since the interest is compounded half-yearly, we need to convert the time into half-year periods:
- Total compounding periods (n) = 1.5 years × 2 = 3 periods

### Step 3: Calculate the rate per compounding period
Since the rate is given per annum, we need to find the rate for each half-year:
- Rate per period (r) = 8% / 2 = 4%

### Step 4: 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

Substituting the values into the formula:
\[ A = 31250 \left(1 + \frac{4}{100}\right)^3 \]
\[ A = 31250 \left(1 + 0.04\right)^3 \]
\[ A = 31250 \left(1.04\right)^3 \]

### Step 5: Calculate \( (1.04)^3 \)
Calculating \( (1.04)^3 \):
\[ (1.04)^3 = 1.124864 \]

### Step 6: Calculate the amount (A)
Now, substituting back to find A:
\[ A = 31250 \times 1.124864 \]
\[ A \approx 35153.00 \]

### Step 7: Calculate the compound interest (CI)
Compound Interest (CI) can be calculated using the formula:
\[ CI = A - P \]
Substituting the values:
\[ CI = 35153.00 - 31250 \]
\[ CI \approx 3903.00 \]

### Final Results
- Amount (A) = Rs 35,153.00
- Compound Interest (CI) = Rs 3,903.00

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