An amount at certain rate of s.i become 5 times itself in 3 years.how much times to take to becomes 45 times itself what is the solution
To solve this problem, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal (initial amount)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time in years
In the given scenario, the interest is compounded once every year (n = 1), and we need to find the time it takes for the principal to become 45 times itself.
Let's assume the initial amount (principal) is P. According to the problem, the amount becomes 5 times the principal after 3 years:
5P = P(1 + r/1)^(1*3)
5 = (1 + r)^3
Now, we can solve for the annual interest rate (r) by taking the cube root of both sides:
(1 + r) = ∛5
r = ∛5 - 1
Next, we need to find the time it takes for the amount to become 45 times the principal:
45P = P(1 + (∛5 - 1)/1)^(1*t)
45 = (∛5)^t
To solve for t, we can take the logarithm of both sides:
log(45) = t * log(∛5)
t = log(45) / log(∛5)
Using a calculator, we can find the value of t:
t ≈ log(45) / log(∛5) ≈ 4.016
Therefore, it would take approximately 4.016 years for the principal amount to become 45 times itself.