In how many years will Rs 2,000 yield Rs 662 as compound interest at 10% per annum compounded annually?

Options

A. 3
B. 2
C. 4
D. 5

Answer

To solve the problem of how many years it will take for Rs 2,000 to yield Rs 662 as compound interest at a rate of 10% per annum compounded annually, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs 2,000
- Compound Interest (CI) = Rs 662
- Rate of interest (R) = 10% per annum

### Step 2: Calculate the Total Amount (A)
The total amount (A) after earning compound interest can be calculated as:
\[ A = P + CI \]
Substituting the values:
\[ A = 2000 + 662 = 2662 \]

### Step 3: Set up the ratio of Amount to Principal
Now we can set up the ratio of the amount (A) to the principal (P):
\[ \text{Ratio} = \frac{A}{P} = \frac{2662}{2000} \]
To simplify this ratio, we can divide both numbers by 2:
\[ \text{Ratio} = \frac{1331}{1000} \]

### Step 4: Relate the ratio to compound interest
We know that the formula for compound interest can be expressed in terms of the ratio of the amount to the principal:
\[ \frac{A}{P} = \left(1 + \frac{R}{100}\right)^n \]
Here, \(R = 10\%\), so:
\[ \frac{A}{P} = \left(1 + \frac{10}{100}\right)^n = \left(1.1\right)^n \]

### Step 5: Set the two ratios equal
Now we can set the two ratios equal to each other:
\[ \frac{1331}{1000} = (1.1)^n \]

### Step 6: Take logarithm to solve for n
To solve for \(n\), we can take the logarithm of both sides:
\[ \log\left(\frac{1331}{1000}\right) = n \cdot \log(1.1) \]
Now, calculate the logarithms:
- \( \log(1331) \approx 3.1249 \)
- \( \log(1000) = 3 \)
- \( \log(1.1) \approx 0.0414 \)

So:
\[ \log\left(\frac{1331}{1000}\right) = 3.1249 - 3 = 0.1249 \]

### Step 7: Solve for n
Now, substituting the values into the equation:
\[ 0.1249 = n \cdot 0.0414 \]
\[ n = \frac{0.1249}{0.0414} \approx 3.02 \]

Since \(n\) must be a whole number, we round it to the nearest whole number, which is 3.

### Conclusion
Thus, it will take approximately **3 years** for Rs 2,000 to yield Rs 662 as compound interest at a rate of 10% per annum compounded annually.

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