An amount is said to triple in 6 years with compound interest. How many years will it take for the amount to grow to 27 times its original value?

The correct answer is 18 (Option 1).

Step-by-step Explanation:

1. Understanding the Compound Interest Formula:

Let the principal amount be _P, the annual interest rate be _r, and the time in years be _t.

The total amount _A after _t years under compound interest is given by:

$A=P{\left(1+r\right)}^t$

2. Analyzing the First Condition:

We are given that the amount triples (becomes 3 times the original principal _P) in 6 years.

Substituting _{A = 3P} and _{t = 6} into the formula:

$3P=P{\left(1+r\right)}^6$

Dividing both sides by _P:

$3={\left(1+r\right)}^6$

3. Analyzing the Second Condition:

We need to find the number of years _T for the amount to grow to 27 times its original value (_{A = 27P}).

$27P=P{\left(1+r\right)}^T$

Dividing both sides by _P:

$27={\left(1+r\right)}^T$

4. Solving for Time (_T):

Notice that 27 can be expressed as a power of 3, specifically:

$27=3^3$

Substitute _{3 = (1 + r)⁶} into the equation above:

$27={\left[{\left(1+r\right)}^6\right]}^3$

Using the exponent rule _{(x^a)^b = x^a×b}:

$27={\left(1+r\right)}^{6\times 3}$

$27={\left(1+r\right)}^{18}$

Equating this to _{27 = (1 + r)^T}, we get:

$T=18$

Therefore, it will take 18 years for the amount to become 27 times its original value.