A sum of money invested at compound interest triples itself in five years.


In how many years will it become 27 times itself at the same rate?


Let's assume the initial sum of money is 'P'.


Compound interest is the interest calculated on both the initial principal and the accumulated interest from previous periods. It is calculated using the formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial sum of money)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years the money is invested/borrowed for


We are given that the initial sum of money triples itself in five years. This means that after five years, the future value (A) is three times the initial sum (P). Mathematically, we can represent this as:
A = 3P


To find the time it takes for the money to become 27 times itself, we need to solve for 't' in the equation 27P = P(1 + r/n)^(nt).

Let's compare the given information and the equation:
A = 3P (given)
27P = P(1 + r/n)^(nt) (equation)

We can see that the future value (A) in the given information is equal to 27P in the equation. Therefore, we can equate the two:
3P = P(1 + r/n)^(nt)
27P = P(1 + r/n)^(nt)


Since we have the same initial sum (P) on both sides of the equation, we can cancel it out:
3 = (1 + r/n)^(nt)
27 = (1 + r/n)^(nt)


We can rewrite 3 as (1 + r/n)^(nt)^(1/2) and 27 as (1 + r/n)^(nt)^(3/2):
(1 + r/n)^(nt)^(1/2) = (1 + r/n)^(nt)^(3/2)

Taking the square of the left side and the cube of the right side, we get:
(1 + r/n)^(nt) = (1 + r/n)^(3nt)

Since the bases are the same, we can equate the exponents:
nt = 3nt

Cancelling out 't' on both sides, we get:
n = 3

Therefore, the time it takes for the money to become 27 times itself at the same rate is 'n' years.


As per the given information, we have n = 3. Hence, the money will become 27 times itself at the same rate in 3 years.

45yrs