A sum put out at 4% compound interest payable half yearly amounts to Rs. 6632.55 in `1(1)/(2)` years. The sum is

Answer

To find the principal amount when a sum is put out at 4% compound interest payable half-yearly and amounts to Rs. 6632.55 in 1.5 years, we can follow these steps:

### Step 1: Understand the parameters
- **Rate of interest (R)**: 4% per annum
- **Time (T)**: 1.5 years
- **Amount (A)**: Rs. 6632.55

### Step 2: Convert the rate and time for half-yearly compounding
Since the interest is compounded half-yearly:
- The half-yearly rate (r) = 4% / 2 = 2% = 0.02 (in decimal)
- The total number of half-year periods (n) = 1.5 years × 2 = 3 half-years

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + r\right)^n \]
Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount (what we want to find)
- \( r \) = Rate of interest per period
- \( n \) = Number of periods

### Step 4: Substitute the known values into the formula
We need to rearrange the formula to solve for \( P \):
\[ P = \frac{A}{\left(1 + r\right)^n} \]

Substituting the values:
\[ P = \frac{6632.55}{\left(1 + 0.02\right)^3} \]
\[ P = \frac{6632.55}{(1.02)^3} \]

### Step 5: Calculate \( (1.02)^3 \)
Calculating \( (1.02)^3 \):
\[ (1.02)^3 = 1.061208 \]

### Step 6: Calculate the principal amount \( P \)
Now substituting back:
\[ P = \frac{6632.55}{1.061208} \]
\[ P \approx 6249.99 \]

Thus, the principal amount is approximately Rs. 6250.

### Summary of the Solution
The principal amount that was put out at compound interest is approximately Rs. 6250.

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