What is the rate of compound interest ?
(i) The principal was invested for 4 years.
(ii) The earned interest was Rs. 1491.

Answer

To determine the rate of compound interest based on the given conditions, we can follow these steps:

### Step 1: Understand the Formula for Compound Interest
The formula for compound interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^N - P \]
Where:
- \( CI \) = Compound Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( N \) = Number of years the money is invested or borrowed

### Step 2: Substitute the Known Values
From the problem, we know:
- The earned interest (CI) is Rs. 1491.
- The principal was invested for \( N = 4 \) years.

We can rewrite the formula for CI as:
\[ 1491 = P \left(1 + \frac{R}{100}\right)^4 - P \]

### Step 3: Rearrange the Equation
Rearranging the equation gives us:
\[ 1491 = P \left[\left(1 + \frac{R}{100}\right)^4 - 1\right] \]

### Step 4: Isolate the Compound Interest Factor
To isolate the compound interest factor, we can express it as:
\[ \frac{1491}{P} = \left(1 + \frac{R}{100}\right)^4 - 1 \]
This means:
\[ \left(1 + \frac{R}{100}\right)^4 = \frac{1491}{P} + 1 \]

### Step 5: Solve for the Rate of Interest
Let \( x = 1 + \frac{R}{100} \). Then we have:
\[ x^4 = \frac{1491}{P} + 1 \]
From this, we can express \( R \):
\[ R = 100(x - 1) \]

### Step 6: Conclusion
Without the value of \( P \) (the principal), we cannot determine a specific numerical value for \( R \). Therefore, we can conclude that while we can express \( R \) in terms of \( P \), we cannot find a unique solution for the rate of compound interest without knowing the principal amount.