In how many years will `Rs 2,000` amounts to `Rs. 2,420` at `10%` per annum compound interest?

Options

A. 3 years
B. `2(1)/2 years`
C. 2 years
D. `1(1)/2 years`

Answer

To find out how many years it will take for Rs. 2000 to amount to Rs. 2420 at a compound interest rate of 10% per annum, we can use the formula for compound interest:

\[ A = P \left(1 + \frac{R}{100}\right)^t \]

Where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest per annum
- \( t \) = Time in years

### Step 1: Identify the values
From the question, we have:
- \( A = 2420 \)
- \( P = 2000 \)
- \( R = 10 \)

### Step 2: Substitute the values into the formula
We substitute the known values into the compound interest formula:

\[ 2420 = 2000 \left(1 + \frac{10}{100}\right)^t \]

### Step 3: Simplify the equation
Calculate \( 1 + \frac{10}{100} \):

\[ 1 + \frac{10}{100} = 1 + 0.1 = 1.1 \]

Now, substitute this back into the equation:

\[ 2420 = 2000 \cdot (1.1)^t \]

### Step 4: Divide both sides by 2000
To isolate \( (1.1)^t \), divide both sides by 2000:

\[ \frac{2420}{2000} = (1.1)^t \]

Calculating the left side:

\[ \frac{2420}{2000} = 1.21 \]

So we have:

\[ 1.21 = (1.1)^t \]

### Step 5: Take logarithm on both sides
To solve for \( t \), we can take the logarithm of both sides:

\[ \log(1.21) = \log((1.1)^t) \]

Using the power rule of logarithms:

\[ \log(1.21) = t \cdot \log(1.1) \]

### Step 6: Solve for \( t \)
Now, isolate \( t \):

\[ t = \frac{\log(1.21)}{\log(1.1)} \]

### Step 7: Calculate the logarithms
Using a calculator, we find:

- \( \log(1.21) \approx 0.0828 \)
- \( \log(1.1) \approx 0.0414 \)

Now substitute these values into the equation:

\[ t = \frac{0.0828}{0.0414} \approx 2 \]

### Conclusion
Thus, it will take approximately **2 years** for Rs. 2000 to amount to Rs. 2420 at a compound interest rate of 10% per annum.

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