A sum of Rs. 10500 becomes Rs. 17745 in 2 years at the rate of compound interest. If the interest is compounded annually then what will be the rate of interest?

Options

A. A) 40%
B. B) 30%
C. C) 25%
D. D) 20%

Answer

To find the rate of compound interest when a sum of Rs. 10,500 becomes Rs. 17,745 in 2 years, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 10,500
- Amount (A) = Rs. 17,745
- Time (n) = 2 years

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

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 17745 = 10500 \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Divide both sides by the principal
To isolate the term with r, divide both sides by 10,500:
\[ \frac{17745}{10500} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Calculate the left side
Calculating the left side:
\[ \frac{17745}{10500} = 1.69 \]

### Step 6: Rewrite the equation
Now we have:
\[ 1.69 = \left(1 + \frac{r}{100}\right)^2 \]

### Step 7: Take the square root of both sides
Taking the square root of both sides:
\[ \sqrt{1.69} = 1 + \frac{r}{100} \]
\[ 1.3 = 1 + \frac{r}{100} \]

### Step 8: Solve for r
Subtract 1 from both sides:
\[ 1.3 - 1 = \frac{r}{100} \]
\[ 0.3 = \frac{r}{100} \]

### Step 9: Multiply by 100 to find r
Multiply both sides by 100:
\[ r = 0.3 \times 100 \]
\[ r = 30 \]

### Conclusion
The rate of interest is **30%**.

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