A sum of Rs 12500 is deposited for `1 (1)/(2)` years, compounded half-yearly. It amounts to 13000 at the end of first half-year. Find :
rate of interest

Answer

To find the rate of interest for the given problem, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs 12,500
- Amount (A) after the first half-year = Rs 13,000
- Time (T) = 1.5 years = 3/2 years
- Since the interest is compounded half-yearly, we need to consider the time in half-year periods.

### Step 2: Calculate the number of compounding periods
Since the interest is compounded half-yearly:
- Total time in half-year periods = \( \frac{3}{2} \) years = 3 half-year periods.

### Step 3: Use the compound interest formula
The formula for compound interest when compounded half-yearly is:
\[ A = P \left(1 + \frac{r}{200}\right)^{2t} \]

Where:
- A = Amount after time t
- P = Principal
- r = Rate of interest per annum
- t = Time in years

### Step 4: Substitute the known values into the formula
We know that at the end of the first half-year, the amount is Rs 13,000. Therefore, we can use the formula for just the first half-year:
\[ 13000 = 12500 \left(1 + \frac{r}{200}\right)^{1} \]

### Step 5: Simplify the equation
Rearranging the equation gives:
\[ 1 + \frac{r}{200} = \frac{13000}{12500} \]

Calculating the right side:
\[ \frac{13000}{12500} = 1.04 \]

So we have:
\[ 1 + \frac{r}{200} = 1.04 \]

### Step 6: Solve for r
Subtract 1 from both sides:
\[ \frac{r}{200} = 1.04 - 1 \]
\[ \frac{r}{200} = 0.04 \]

Now, multiply both sides by 200 to find r:
\[ r = 0.04 \times 200 \]
\[ r = 8 \]

### Conclusion
The rate of interest is **8%**.

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