A sum of money becomes Rs. 3600 in 2 years and Rs. 5184 in 4 years on compound interest. Find the rate of compound interest.

Answer

To find the rate of compound interest given that a sum of money becomes Rs. 3600 in 2 years and Rs. 5184 in 4 years, we can follow these steps:

### Step 1: Identify the given values
- Amount after 2 years (A1) = Rs. 3600
- Amount after 4 years (A2) = Rs. 5184
- Time for A1 (T1) = 2 years
- Time for A2 (T2) = 4 years

### Step 2: Write the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- A = Amount after time T
- P = Principal amount (initial sum)
- R = Rate of interest
- T = Time in years

### Step 3: Set up the equations
For the first case (after 2 years):
\[ 3600 = P \left(1 + \frac{R}{100}\right)^2 \] (Equation 1)

For the second case (after 4 years):
\[ 5184 = P \left(1 + \frac{R}{100}\right)^4 \] (Equation 2)

### Step 4: Divide Equation 2 by Equation 1
Dividing Equation 2 by Equation 1:
\[ \frac{5184}{3600} = \frac{P \left(1 + \frac{R}{100}\right)^4}{P \left(1 + \frac{R}{100}\right)^2} \]

This simplifies to:
\[ \frac{5184}{3600} = \left(1 + \frac{R}{100}\right)^{4-2} \]
\[ \frac{5184}{3600} = \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Calculate the left-hand side
Calculating \( \frac{5184}{3600} \):
\[ \frac{5184}{3600} = 1.44 \]

### Step 6: Set up the equation
Now we have:
\[ 1.44 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 7: Take the square root
Taking the square root of both sides:
\[ \sqrt{1.44} = 1 + \frac{R}{100} \]
\[ 1.2 = 1 + \frac{R}{100} \]

### Step 8: Solve for R
Subtracting 1 from both sides:
\[ 1.2 - 1 = \frac{R}{100} \]
\[ 0.2 = \frac{R}{100} \]

Multiplying both sides by 100:
\[ R = 20 \]

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

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