Find the difference between compound interest and simple interest on Rs 12,000 and in `1(1)/(2)` years at 10% compounded half-yealry.

Answer

To find the difference between compound interest (CI) and simple interest (SI) on Rs 12,000 over 1.5 years at a rate of 10% compounded half-yearly, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs 12,000
- Rate of interest (R) = 10% per annum
- Time (T) = 1.5 years

### Step 2: Calculate Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{12000 \times 10 \times \frac{3}{2}}{100} \]
Calculating this:
\[ SI = \frac{12000 \times 10 \times 1.5}{100} = \frac{180000}{100} = 1800 \]
So, the Simple Interest (SI) is Rs 1800.

### Step 3: Calculate Compound Interest (CI)
Since the interest is compounded half-yearly, we need to adjust the rate and time:
- Half-yearly rate (R/2) = 10% / 2 = 5%
- Number of compounding periods (n) = 1.5 years × 2 = 3 periods

The formula for Compound Interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where A is the amount after time T. Substituting the values:
\[ A = 12000 \left(1 + \frac{5}{100}\right)^3 \]
Calculating this step-by-step:
\[ A = 12000 \left(1 + 0.05\right)^3 = 12000 \left(1.05\right)^3 \]
Calculating \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
Now substituting back:
\[ A = 12000 \times 1.157625 = 13891.5 \]
So, the total amount (A) after 1.5 years is Rs 13891.5.

Now, we can find the Compound Interest (CI):
\[ CI = A - P = 13891.5 - 12000 = 1891.5 \]
So, the Compound Interest (CI) is Rs 1891.5.

### Step 4: Find the difference between CI and SI
Now, we can find the difference:
\[ Difference = CI - SI = 1891.5 - 1800 = 91.5 \]
Thus, the difference between Compound Interest and Simple Interest is Rs 91.5.

### Summary of Results
- Simple Interest (SI) = Rs 1800
- Compound Interest (CI) = Rs 1891.5
- Difference = Rs 91.5