An initial capital deposit grows to twice its original value in 8 years when invested at a constant rate of simple interest. How many years will it take for this capital to become four times its initial value at the same interest rate?

The correct option is 24 years.

Let the initial principal deposit be $P$ and the constant rate of simple interest per annum be $R$.

Step 1: Calculate the interest rate ($R$)

When the capital deposit grows to twice its original value in 8 years ($T_1=8$ years), the total amount becomes:

$A_1=2P$

The simple interest earned ($I_1$) is:

$I_1=A_1-P=2P-P=P$

Using the simple interest formula $I=\frac{P\times R\times T}{100}$:

$P=\frac{P\times R\times 8}{100}$

Dividing both sides by $P$:

$1=\frac{8R}{100}$

$R=\frac{100}{8}=12.5\%\text{per annum}$

Step 2: Calculate the time required ($T_2$) to become four times the initial value

When the capital deposit grows to four times its original value, the total amount is:

$A_2=4P$

The simple interest required ($I_2$) is:

$I_2=A_2-P=4P-P=3P$

Using the simple interest formula with time $T_2$:

$3P=\frac{P\times R\times T_2}{100}$

Substituting $R=\frac{100}{8}$:

$3P=\frac{P\times \left(\frac{100}{8}\right)\times T_2}{100}$

Simplifying the right side:

$3P=\frac{P\times T_2}{8}$

Dividing both sides by $P$:

$3=\frac{T_2}{8}$

Solving for $T_2$:

$T_2=3\times 8=24\text{years}$

Thus, it will take 24 years for the capital to become four times its initial value at the same interest rate.