If 30000 becomes 39930 in 1.5 years at certain rate of interest then find the rate of interest if interest is compounded semi-annually?

Options

A. `25%`
B. `20%`
C. `15%`
D. `30%`

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) = 30,000
- Amount (A) = 39,930
- Time (T) = 1.5 years
- Compounding frequency = Semi-annually

### Step 2: Convert the time to the number of compounding periods
Since the interest is compounded semi-annually, we need to convert the time from years to the number of 6-month periods:
- Number of compounding periods (n) = 1.5 years × 2 = 3 periods

### Step 3: Use the formula for compound interest
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- A = Amount after time t
- P = Principal amount
- r = Annual interest rate (as a decimal)
- n = Number of times interest is compounded per year
- t = Time in years

Substituting the known values into the formula:
\[ 39,930 = 30,000 \left(1 + \frac{r}{2}\right)^{3} \]

### Step 4: Simplify the equation
First, divide both sides by 30,000:
\[ \frac{39,930}{30,000} = \left(1 + \frac{r}{2}\right)^{3} \]
\[ 1.331 = \left(1 + \frac{r}{2}\right)^{3} \]

### Step 5: Take the cube root of both sides
To isolate the term with r, take the cube root:
\[ 1 + \frac{r}{2} = 1.331^{\frac{1}{3}} \]

Calculating the cube root of 1.331:
\[ 1 + \frac{r}{2} \approx 1.1 \]

### Step 6: Solve for r
Now, isolate r:
\[ \frac{r}{2} = 1.1 - 1 \]
\[ \frac{r}{2} = 0.1 \]
\[ r = 0.2 \]

### Step 7: Convert r to a percentage
To express r as a percentage, multiply by 100:
\[ r = 0.2 \times 100 = 20\% \]

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

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