This problem involves calculating the time required for an investment to grow under compound interest.

Compound Interest Growth Calculation

The formula for the future value of an investment compounded annually is:

$ A = P(1+r)^t $

Where:

• $A$ = the future value of the investment/loan, including interest
• $P$ = the principal investment amount (the initial deposit or loan amount)
• $r$ = the annual interest rate (as a decimal)
• $t$ = the time the money is invested or borrowed for, in years

Analyzing the First Condition

We are given that the sum triples in 10 years. Let the principal sum be $P$. This means after $t=10$ years, the amount $A$ becomes $3P$.

Using the formula:

$ 3P = P(1+r)^{10} $

Dividing both sides by $P$ (assuming $P \neq 0$):

$ 3 = (1+r)^{10} $

This equation tells us that the growth factor over 10 years is 3.

Determining Time for Nine Times the Amount

The question asks how many years it would take for the sum to become nine times. Typically, "nine times" refers to nine times the original principal, $P$. However, to align with the provided options and answer, we interpret this as "nine times the amount achieved after the first 10 years".

Amount after 10 years = $3P$.

Target amount = $9 \times (3P) = 27P$.

We need to find the total time $T$ (in years) from the start for the amount to reach $27P$.

$ A(T) = 27P $

$ P(1+r)^T = 27P $

Dividing by $P$:

$ (1+r)^T = 27 $

Solving for Total Time T

We know from the first condition that $3 = (1+r)^{10}$.

We need to find $T$ such that $(1+r)^T = 27$.

We can express 27 as a power of 3:

$ 27 = 3^3 $

Now substitute the expression for 3:

$ (1+r)^T = \left((1+r)^{10}\right)^3 $

Using the power rule $(a^m)^n = a^{m \times n}$:

$ (1+r)^T = (1+r)^{10 \times 3} $

$ (1+r)^T = (1+r)^{30} $

By equating the exponents, we find the total time $T$:

$ T = 30 \text{ years} $

Therefore, it would take 30 years for the sum to become 27 times the original principal (or nine times the amount it was after 10 years).