After how many years, will Rs. 1,728 become Rs.2,197 at `8(1)/(3)%` p.a. compound interest?

Options

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

Answer

To find out how many years it will take for Rs. 1,728 to become Rs. 2,197 at an interest rate of \(8 \frac{1}{3}\%\) per annum compounded annually, we can follow these steps:

### Step 1: Convert the interest rate to a fraction
The interest rate given is \(8 \frac{1}{3}\%\). We can convert this to an improper fraction:
\[ 8 \frac{1}{3} = \frac{25}{3}\% \]
To convert this percentage to a decimal for calculations, we divide by 100:
\[ \frac{25}{3} \div 100 = \frac{25}{300} = \frac{1}{12} \]

### Step 2: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) is the final amount (Rs. 2,197)
- \(P\) is the principal amount (Rs. 1,728)
- \(r\) is the rate of interest (which we found to be \(\frac{1}{12}\))
- \(n\) is the number of years

Substituting the known values into the formula:
\[ 2197 = 1728 \left(1 + \frac{1}{12}\right)^n \]

### Step 3: Simplify the equation
We can simplify \(1 + \frac{1}{12}\):
\[ 1 + \frac{1}{12} = \frac{12 + 1}{12} = \frac{13}{12} \]
Now, substituting this back into the equation gives:
\[ 2197 = 1728 \left(\frac{13}{12}\right)^n \]

### Step 4: Divide both sides by 1728
To isolate the term with \(n\):
\[ \frac{2197}{1728} = \left(\frac{13}{12}\right)^n \]

### Step 5: Calculate \(\frac{2197}{1728}\)
Calculating the left side:
\[ \frac{2197}{1728} = \left(\frac{13}{12}\right)^n \]
We can recognize that:
\[ 2197 = 13^3 \quad \text{and} \quad 1728 = 12^3 \]
Thus:
\[ \frac{2197}{1728} = \frac{13^3}{12^3} = \left(\frac{13}{12}\right)^3 \]

### Step 6: Set the exponents equal
Since both sides are equal:
\[ \left(\frac{13}{12}\right)^n = \left(\frac{13}{12}\right)^3 \]
This implies:
\[ n = 3 \]

### Conclusion
Therefore, it will take **3 years** for Rs. 1,728 to become Rs. 2,197 at an interest rate of \(8 \frac{1}{3}\%\) per annum compounded annually.

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