At what rate per annum will Rs. 32000 yield a compound interest of Rs. 5044 in 9 months Interest being compounded quarterly?

Options

A. `20% `
B. `32% `
C. `50% `
D. `80%`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Identify the Given Values
- Principal (P) = Rs. 32,000
- Compound Interest (CI) = Rs. 5,044
- Time (T) = 9 months = 9/12 years = 0.75 years
- Since the interest is compounded quarterly, the number of compounding periods in 9 months = 3 (as there are 4 quarters in a year).

### Step 2: Calculate the Total Amount (A)
The total amount (A) after the interest is added can be calculated using the formula:
\[ A = P + CI \]
\[ A = 32,000 + 5,044 = 37,044 \]

### Step 3: Use the Compound Interest Formula
The compound interest formula 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 (in decimal)
- \( n \) = Number of times interest applied per time period
- \( t \) = Time the money is invested for (in years)

### Step 4: Substitute the Known Values
In our case:
- \( A = 37,044 \)
- \( P = 32,000 \)
- \( n = 4 \) (quarterly)
- \( t = 0.75 \)

Substituting these values into the formula:
\[ 37,044 = 32,000 \left(1 + \frac{r}{4}\right)^{4 \times 0.75} \]
\[ 37,044 = 32,000 \left(1 + \frac{r}{4}\right)^{3} \]

### Step 5: Simplify the Equation
Divide both sides by 32,000:
\[ \frac{37,044}{32,000} = \left(1 + \frac{r}{4}\right)^{3} \]
Calculating the left side:
\[ 1.1576875 = \left(1 + \frac{r}{4}\right)^{3} \]

### Step 6: Take the Cube Root
To solve for \( 1 + \frac{r}{4} \), take the cube root of both sides:
\[ 1 + \frac{r}{4} = (1.1576875)^{\frac{1}{3}} \]
Calculating the cube root:
\[ 1 + \frac{r}{4} \approx 1.050 \]

### Step 7: Solve for \( r \)
Now, isolate \( r \):
\[ \frac{r}{4} = 1.050 - 1 \]
\[ \frac{r}{4} = 0.050 \]
Multiply by 4:
\[ r = 0.050 \times 4 = 0.20 \]

### Step 8: Convert to Percentage
Convert \( r \) to a percentage:
\[ r = 0.20 \times 100 = 20\% \]

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

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