A sum of money at compound interest amounts to Rs. 4050 in one year and to Rs. 4723.92 in 3 years. Find the rate of interest per annum.

Answer

To find the rate of interest per annum when a sum of money amounts to Rs. 4050 in one year and Rs. 4723.92 in three years at compound interest, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The amount \( A \) after \( n \) years at compound interest can be calculated using the formula:
\[ A = P(1 + r)^n \]
where:
- \( A \) = Amount after \( n \) years
- \( P \) = Principal amount (initial sum of money)
- \( r \) = Rate of interest per annum (in decimal)
- \( n \) = Number of years

### Step 2: Set Up the Equations
From the problem, we have two amounts:
1. After 1 year:
\[ A_1 = P(1 + r) = 4050 \]
2. After 3 years:
\[ A_3 = P(1 + r)^3 = 4723.92 \]

### Step 3: Express \( P \) in Terms of \( r \)
From the first equation, we can express \( P \):
\[ P = \frac{4050}{1 + r} \]

### Step 4: Substitute \( P \) in the Second Equation
Now, substitute \( P \) into the second equation:
\[ 4723.92 = \left(\frac{4050}{1 + r}\right)(1 + r)^3 \]
This simplifies to:
\[ 4723.92 = 4050(1 + r)^2 \]

### Step 5: Solve for \( (1 + r)^2 \)
Rearranging gives:
\[ (1 + r)^2 = \frac{4723.92}{4050} \]
Calculating the right side:
\[ (1 + r)^2 = 1.165 \]

### Step 6: Take the Square Root
Now take the square root of both sides:
\[ 1 + r = \sqrt{1.165} \]
Calculating the square root:
\[ 1 + r \approx 1.080 \]

### Step 7: Solve for \( r \)
Now, subtract 1 from both sides:
\[ r \approx 1.080 - 1 = 0.080 \]
Convert \( r \) to a percentage:
\[ r \approx 0.080 \times 100 = 8\% \]

### Conclusion
The rate of interest per annum is approximately **8%**.

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