A sum of money amounts Rs. 800 in 3 years and Rs. 840 in 4 years at compound interest. What is the rate of compound interest per annum?

Answer

To find the rate of compound interest per annum, we can follow these steps:

### Step 1: Identify the given amounts and time periods
We have:
- Amount after 3 years (A1) = Rs. 800
- Amount after 4 years (A2) = Rs. 840

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

### Step 3: Set up equations for both time periods
For the first case (3 years):
\[ 800 = P \left(1 + \frac{R}{100}\right)^3 \] (Equation 1)

For the second case (4 years):
\[ 840 = P \left(1 + \frac{R}{100}\right)^4 \] (Equation 2)

### Step 4: Divide Equation 2 by Equation 1
To eliminate \( P \), we divide the second equation by the first:
\[ \frac{840}{800} = \frac{P \left(1 + \frac{R}{100}\right)^4}{P \left(1 + \frac{R}{100}\right)^3} \]
This simplifies to:
\[ \frac{840}{800} = 1 + \frac{R}{100} \]

### Step 5: Simplify the left-hand side
Calculating \( \frac{840}{800} \):
\[ \frac{840}{800} = \frac{84}{80} = 1.05 \]
So we have:
\[ 1.05 = 1 + \frac{R}{100} \]

### Step 6: Solve for \( R \)
Subtract 1 from both sides:
\[ 1.05 - 1 = \frac{R}{100} \]
\[ 0.05 = \frac{R}{100} \]
Now, multiply both sides by 100:
\[ R = 0.05 \times 100 = 5 \]

### Conclusion
The rate of compound interest per annum is:
\[ \boxed{5\%} \]

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