Consider a $100 investment earning 12% compounded quarterly. How long will it take to triple your money? Round your answer to the nearest tenth of a year.
(1+.12/4)^(4x) = 3
x = 9.3 years
Makes sense. The Rule of 72 says it will double in about 6 years.
To answer this question, we need to use the compound interest formula:
A = P * (1 + r/n)^(nt)
Where:
A is the final amount after time t
P is the principal amount (initial investment)
r is the annual interest rate (as a decimal)
n is the number of times interest is compounded per year
t is the time in years
In this case, the principal amount is $100, the annual interest rate is 12%, compounded quarterly (n = 4), and we want to find the time it takes for the investment to triple (A = 3P).
Let's solve the equation for t:
3P = P * (1 + r/n)^(nt)
Divide both sides of the equation by P:
3 = (1 + r/n)^(nt)
Take the natural logarithm of both sides to isolate nt:
ln(3) = nt * ln(1 + r/n)
We can now solve for t. Rearrange the equation:
t = ln(3) / (n * ln(1 + r/n))
Substituting the given values:
t = ln(3) / (4 * ln(1 + 0.12/4))
Calculating:
t ≈ ln(3) / (4 * ln(1.03))
t ≈ ln(3) / (4 * 0.0307)
t ≈ ln(3) / 0.1229
t ≈ 5.66 (rounded to two decimal places)
Therefore, it will take approximately 5.7 years (rounded to the nearest tenth of a year) to triple your money with a 12% annual interest rate compounded quarterly.