At what rate will a sum of Rs. 1000 amounts to Rs. 1102.50 in 2 years at compound interest?

Options

A. A)0.065
B. B)0.06
C. C)0.05
D. D)0.055

Answer

To find the rate at which a sum of Rs. 1000 amounts to Rs. 1102.50 in 2 years at compound interest, we can follow these steps:

### Step 1: Identify the Principal and Amount
The principal (P) is Rs. 1000 and the amount (A) after 2 years is Rs. 1102.50.

### Step 2: Set Up the Formula for Compound Interest
The formula for the amount in compound interest is given by:
\[ A = P(1 + r)^n \]
where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal amount
- \( r \) = Rate of interest (in decimal)
- \( n \) = Number of years

### Step 3: Substitute the Known Values
Substituting the known values into the formula:
\[ 1102.50 = 1000(1 + r)^2 \]

### Step 4: Simplify the Equation
Dividing both sides by 1000:
\[ 1.1025 = (1 + r)^2 \]

### Step 5: Take the Square Root
To find \( 1 + r \), take the square root of both sides:
\[ 1 + r = \sqrt{1.1025} \]

### Step 6: Calculate the Square Root
Calculating the square root:
\[ 1 + r = 1.05 \]

### Step 7: Solve for \( r \)
Now, subtract 1 from both sides to find \( r \):
\[ r = 1.05 - 1 = 0.05 \]

### Step 8: Convert to Percentage
To convert \( r \) into a percentage, multiply by 100:
\[ r = 0.05 \times 100 = 5\% \]

### Conclusion
The rate at which the sum of Rs. 1000 amounts to Rs. 1102.50 in 2 years at compound interest is **5%**.

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