The compound interest on a sum of 4000 becomes 630.50 in 9 months. Find the rate of interest, if interest is compounded quarterly.

Options

A. 20
B. 23
C. 19
D. 21

Answer

To solve the problem of finding the rate of interest when the compound interest on a sum of ₹4000 becomes ₹630.50 in 9 months, compounded quarterly, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = ₹4000
- Compound Interest (CI) = ₹630.50
- Time (T) = 9 months = 9/12 years = 0.75 years
- Since the interest is compounded quarterly, the number of compounding periods per year (n) = 4.

### Step 2: Calculate the total amount (A)
The total amount (A) after the interest is added can be calculated as:
\[ A = P + CI \]
\[ A = 4000 + 630.50 = 4630.50 \]

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

### Step 4: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 4630.50 = 4000 \left(1 + \frac{r}{4}\right)^{4 \times 0.75} \]
\[ 4630.50 = 4000 \left(1 + \frac{r}{4}\right)^{3} \]

### Step 5: Divide both sides by 4000
\[ \frac{4630.50}{4000} = \left(1 + \frac{r}{4}\right)^{3} \]
\[ 1.157625 = \left(1 + \frac{r}{4}\right)^{3} \]

### Step 6: Take the cube root of both sides
To isolate \(1 + \frac{r}{4}\), take the cube root:
\[ 1 + \frac{r}{4} = (1.157625)^{\frac{1}{3}} \]
Calculating the cube root:
\[ 1 + \frac{r}{4} \approx 1.050 \]

### Step 7: Solve for \(r\)
Subtract 1 from both sides:
\[ \frac{r}{4} \approx 1.050 - 1 \]
\[ \frac{r}{4} \approx 0.050 \]
Now, multiply both sides by 4 to find \(r\):
\[ r \approx 0.050 \times 4 = 0.20 \]

### Step 8: Convert to percentage
To convert \(r\) to a percentage, multiply by 100:
\[ r \approx 0.20 \times 100 = 20\% \]

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

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