At what rate per cent per annum will Rs 640 amount to 774.40 in 2 years when compounded annually?

Answer

To solve the problem step by step, we need to find the rate of interest per annum at which Rs 640 will amount to Rs 774.40 in 2 years when compounded annually.

### Step 1: Identify the given values
- Principal (P) = Rs 640
- Amount (A) = Rs 774.40
- Time (T) = 2 years

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

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 774.40 = 640 \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Simplify the equation
First, divide both sides by 640:
\[ \frac{774.40}{640} = \left(1 + \frac{r}{100}\right)^2 \]
Calculating the left side:
\[ \frac{774.40}{640} = 1.21 \]
So we have:
\[ 1.21 = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Take the square root of both sides
Taking the square root gives:
\[ \sqrt{1.21} = 1 + \frac{r}{100} \]
Calculating the square root:
\[ 1.1 = 1 + \frac{r}{100} \]

### Step 6: Isolate \( r \)
Subtract 1 from both sides:
\[ 1.1 - 1 = \frac{r}{100} \]
This simplifies to:
\[ 0.1 = \frac{r}{100} \]

### Step 7: Solve for \( r \)
Multiply both sides by 100:
\[ r = 0.1 \times 100 \]
Thus:
\[ r = 10 \]

### Conclusion
The rate of interest per annum is **10%**.

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