Find out how long it takes a $3,100 investment to double if it is invested at 7% compounded monthly. Round to the nearest tenth of a year.
To find out how long it takes for a $3,100 investment to double at an interest rate of 7% compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (in this case, twice the initial value, $3,100 * 2 = $6,200)
P = the principal investment ($3,100)
r = annual interest rate (7% or 0.07)
n = number of compounding periods per year (12 since it's compounded monthly)
t = time in years
We need to solve for t, so let's rearrange the formula:
A/P = (1 + r/n)^(nt)
Substituting the given values:
(6200/3100) = (1 + 0.07/12)^(12t)
2 = (1 + 0.00583)^(12t)
Now, we can take the logarithm of both sides. Let's use the natural logarithm:
ln(2) = ln((1 + 0.00583)^(12t))
Using the logarithmic property ln(a^b) = b * ln(a):
ln(2) = 12t * ln(1 + 0.00583)
Now, we can isolate t by dividing both sides by 12 * ln(1 + 0.00583):
t = ln(2) / (12 * ln(1 + 0.00583))
Using a scientific calculator or math software, we can calculate t:
t ≈ 9.5 years
Therefore, it takes approximately 9.5 years for the $3,100 investment to double at a 7% interest rate compounded monthly.
To find out how long it takes for a $3,100 investment to double at a 7% interest rate compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal (initial investment)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = time (in years)
In this case:
P = $3,100
r = 7% = 0.07 (in decimal form)
n = 12 (compounded monthly)
We need to find the value of t that makes A equal to 2P:
2P = P(1 + r/n)^(nt)
Double-checking the equation, we have:
2(3100) = 3100(1 + 0.07/12)^(12t)
Simplifying further:
2 = (1 + 0.00583)^(12t)
Taking the natural logarithm (ln) on both sides:
ln(2) = ln[(1 + 0.00583)^(12t)]
Using the property of logarithms (ln(a^b) = b*ln(a)):
ln(2) = 12t * ln(1.00583)
Now, we can solve for t by dividing both sides by 12 and then dividing further by ln(1.00583):
t = [ln(2)] / [12 * ln(1.00583)]
Calculating this using a calculator, we find:
t ≈ 10.24 years
Therefore, it takes approximately 10.24 years for the investment to double. Rounded to the nearest tenth of a year, it would be 10.2 years.
2 = [1 + (.07 / 12)]^t ... t is in months
log(2) = t log[1 + (.07 / 12)]