Ashima took a loan of Rs 1,00,000 at 12% p.a. compounded half- yearly. She paid Rs 1,12,360. If `(1.06)^(2)` is equal to 1.1236, then the period for which she took the loan is

Options

A. 2 years
B. 1 year
C. 6 months
D. `1(1)/(2)` years

Answer

To solve the problem step by step, we will use the formula for compound interest and the information provided in the question.

### Step 1: Identify the given values
- Principal (P) = Rs. 1,00,000
- Rate of interest (R) = 12% per annum
- Amount (A) paid = Rs. 1,12,360
- Compounding frequency = half-yearly

### Step 2: Convert the annual interest rate to half-yearly
Since the interest is compounded half-yearly, we need to divide the annual interest rate by 2:
\[ \text{Half-yearly Rate} = \frac{12\%}{2} = 6\% \]

### Step 3: Write the formula for compound interest
The formula for the amount (A) in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^{n} \]
where \( n \) is the total number of compounding periods.

### Step 4: Determine the number of compounding periods
Since the interest is compounded half-yearly, if the loan is for \( n \) years, the number of compounding periods will be:
\[ \text{Total compounding periods} = 2n \]

### Step 5: Substitute the values into the formula
Now we can substitute the values into the formula:
\[ 1,12,360 = 1,00,000 \left(1 + \frac{6}{100}\right)^{2n} \]
This simplifies to:
\[ 1,12,360 = 1,00,000 \left(1.06\right)^{2n} \]

### Step 6: Divide both sides by the principal
To isolate the term with \( n \), divide both sides by 1,00,000:
\[ \frac{1,12,360}{1,00,000} = \left(1.06\right)^{2n} \]
This gives us:
\[ 1.1236 = \left(1.06\right)^{2n} \]

### Step 7: Use the given information
We are given that \( (1.06)^{2} = 1.1236 \). Therefore, we can equate the exponents:
\[ 2n = 2 \]

### Step 8: Solve for \( n \)
Now, divide both sides by 2:
\[ n = 1 \]

### Conclusion
The period for which Ashima took the loan is **1 year**.

---