How long (in years) will it take your money to triple at an annual percentage rate of 5% compounded annually? Use logarithms to solve. Round and show 2 decimal places.
3 = 1.05^n
ln3 = n ln1.05
n = ln3/ln1.05 = 22.52 years
To determine how long it takes for money to triple at an annual percentage rate of 5% compounded annually, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, we want to find the value of t. Given that we start with the principal amount P and want to triple it, the future value A will be 3P. The annual interest rate r is 5% or 0.05, and the interest is compounded annually, so n = 1. Substituting these values into the formula:
3P = P(1 + 0.05/1)^(1 * t)
Simplifying the equation:
3 = (1.05)^t
Taking the logarithm of both sides:
log(3) = log(1.05^t)
Using the logarithm power rule (log(a^b) = b * log(a)):
log(3) = t * log(1.05)
Solving for t:
t = log(3) / log(1.05)
Using a calculator, we can evaluate this expression:
t ≈ log(3) / log(1.05) ≈ 22.48
Therefore, it will take approximately 22.48 years for the money to triple at an annual percentage rate of 5% compounded annually. Rounding to two decimal places, the answer is 22.48 years.
To find out how long it will take for the money to triple, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = final amount (which in this case is three times the initial amount)
P = initial principal (money at the beginning)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case, we want A/P to be equal to 3, since we want the money to triple. So we have:
3 = (1 + 0.05/1)^(1*t)
Now we can solve for t using logarithms. Taking the logarithm (base 10 or natural logarithm, both will work) of both sides, we get:
log(3) = log((1 + 0.05/1)^(1*t))
Since the base of the logarithm is the same on both sides, we can use the logarithmic property:
log(a^b) = b * log(a)
Therefore, we have:
log(3) = t * log(1.05)
Now we can solve for t by dividing both sides by log(1.05):
t = log(3) / log(1.05)
Using a calculator, we can find that log(3) is approximately 0.4771 and log(1.05) is approximately 0.0212. Dividing log(3) by log(1.05), we get:
t ≈ 0.4771 / 0.0212 ≈ 22.55
So it will take approximately 22.55 years for the money to triple at an annual interest rate of 5% compounded annually.