At what rate per cent per annum will a sum of Rs 6250 amount to Rs 7840 in 2 years, compounded annually?

Answer

To find the rate per cent per annum at which a sum of Rs 6250 will amount to Rs 7840 in 2 years, compounded annually, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs 6250
- Amount (A) = Rs 7840
- Time (n) = 2 years

### Step 2: Use the formula for compound interest
The formula for the amount (A) in compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( r \) is the rate of interest per annum.

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 7840 = 6250 \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Divide both sides by the principal
To isolate the term with \( r \), divide both sides by 6250:
\[ \frac{7840}{6250} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Simplify the left side
Calculating \( \frac{7840}{6250} \):
\[ \frac{7840}{6250} = 1.2544 \]
So we have:
\[ 1.2544 = \left(1 + \frac{r}{100}\right)^2 \]

### Step 6: Take the square root of both sides
Taking the square root to solve for \( 1 + \frac{r}{100} \):
\[ \sqrt{1.2544} = 1 + \frac{r}{100} \]
Calculating the square root:
\[ 1.12 = 1 + \frac{r}{100} \]

### Step 7: Isolate \( r \)
Now, subtract 1 from both sides:
\[ 1.12 - 1 = \frac{r}{100} \]
\[ 0.12 = \frac{r}{100} \]

### Step 8: Multiply by 100 to find \( r \)
To find \( r \), multiply both sides by 100:
\[ r = 0.12 \times 100 \]
\[ r = 12 \]

### Conclusion
The rate of interest per annum is:
\[ \boxed{12\%} \]

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