An initial capital of ₹1,25,000 is deposited into an investment account earning an annual compound interest rate of 12%. Calculate the duration in years required for this principal to accumulate to a total amount of ₹1,75,616.

The correct option is 3.

To find the duration in years required for the initial capital to accumulate to the total amount, we can use the formula for compound interest:

$A=P{\left(1+\frac{r}{100}\right)}^n$

Where:
$A$ = Total accumulated amount = ₹1,75,616
$P$ = Initial principal amount = ₹1,25,000
$r$ = Annual interest rate = 12%
$n$ = Duration in years

Substitute the given values into the formula:

$\mathrm{1,75,616}=\mathrm{1,25,000}{\left(1+\frac{12}{100}\right)}^n$

Simplify the fraction inside the parentheses:

$1+\frac{12}{100}=1+0.12=1.12=\frac{112}{100}=\frac{28}{25}$

Substitute this back into the equation:

$\mathrm{1,75,616}=\mathrm{1,25,000}{\left(\frac{28}{25}\right)}^n$

Divide both sides by 1,25,000:

$\frac{\mathrm{1,75,616}}{\mathrm{1,25,000}}={\left(\frac{28}{25}\right)}^n$

Simplify the fraction on the left-hand side by dividing both numerator and denominator by 8:

$\frac{\mathrm{1,75,616}÷8}{\mathrm{1,25,000}÷8}=\frac{\mathrm{21,952}}{\mathrm{15,625}}$

Now, rewrite the fraction in terms of powers of 28 and 25:

${25}^3=\mathrm{15,625}$

${28}^3=\mathrm{21,952}$

Thus, we have:

${\left(\frac{28}{25}\right)}^3={\left(\frac{28}{25}\right)}^n$

Comparing the exponents on both sides, we get:

$n=3$

Therefore, the duration required for the investment is 3 years.