Nagma invested Rs. 6000 in a company at compound interest compounded semi-annually. She receives Rs. 7986 after 18 months from the company, the rate of interest per annum.is

Options

A. `10%`
B. `12%`
C. `11%`
D. `20%`

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. 6000
- Amount (A) = Rs. 7986
- Time (T) = 18 months = 18/12 years = 1.5 years
- Compounding frequency (n) = 2 (since it is compounded semi-annually)

### Step 2: Write the formula for compound interest
The formula for the amount A in compound interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- A = final amount
- P = principal amount
- r = rate of interest per annum (in decimal)
- n = number of times interest is compounded per year
- t = time in years

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula gives:
\[ 7986 = 6000 \left(1 + \frac{r}{2}\right)^{2 \times 1.5} \]
This simplifies to:
\[ 7986 = 6000 \left(1 + \frac{r}{2}\right)^{3} \]

### Step 4: Divide both sides by 6000
To isolate the term with r, we divide both sides by 6000:
\[ \frac{7986}{6000} = \left(1 + \frac{r}{2}\right)^{3} \]
Calculating the left side:
\[ 1.331 = \left(1 + \frac{r}{2}\right)^{3} \]

### Step 5: Take the cube root of both sides
To eliminate the exponent, we take the cube root of both sides:
\[ \sqrt[3]{1.331} = 1 + \frac{r}{2} \]
Calculating the cube root:
\[ 1.1 \approx 1 + \frac{r}{2} \]

### Step 6: Solve for r
Now, we solve for r:
\[ 1.1 - 1 = \frac{r}{2} \]
\[ 0.1 = \frac{r}{2} \]
Multiplying both sides by 2:
\[ r = 0.2 \]

### Step 7: Convert r to percentage
To convert r from decimal to percentage, we multiply by 100:
\[ r = 0.2 \times 100 = 20\% \]

### Conclusion
The rate of interest per annum is **20%**.

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