At what rate per cent per annum will ₹ 32000 yield a compound interest of ₹ 5044 in 9 months interest being compounded quarterly

Answer

To find the rate of interest per annum at which ₹32,000 will yield a compound interest of ₹5,044 in 9 months, compounded quarterly, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = ₹32,000
- Compound Interest (CI) = ₹5,044
- Time (T) = 9 months = 9/12 years = 3/4 years
- Compounding frequency = quarterly = 4 times a year

### Step 2: Calculate the total amount (A)
The total amount (A) after the interest is added can be calculated using the formula:
\[ A = P + CI \]
Substituting the values:
\[ A = 32000 + 5044 = 37044 \]

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

Substituting the known values:
\[ 37044 = 32000 \left(1 + \frac{R}{4}\right)^{4 \times \frac{3}{4}} \]
This simplifies to:
\[ 37044 = 32000 \left(1 + \frac{R}{4}\right)^{3} \]

### Step 4: Divide both sides by the principal
To isolate the term with R, divide both sides by 32,000:
\[ \frac{37044}{32000} = \left(1 + \frac{R}{4}\right)^{3} \]

Calculating the left side:
\[ \frac{37044}{32000} = 1.157625 \]

### Step 5: Take the cube root
To solve for \(1 + \frac{R}{4}\), take the cube root of both sides:
\[ 1 + \frac{R}{4} = \sqrt[3]{1.157625} \]

Calculating the cube root:
\[ 1 + \frac{R}{4} \approx 1.05 \]

### Step 6: Solve for R
Now, isolate R:
\[ \frac{R}{4} = 1.05 - 1 \]
\[ \frac{R}{4} = 0.05 \]
Multiplying both sides by 4:
\[ R = 0.20 \]

### Step 7: Convert R to percentage
To express R as a percentage:
\[ R = 0.20 \times 100 = 20\% \]

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

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