How long will it take $1400 to double at 12% annual interest compounded quarterly?
you want (1+.12/4)^n = 2
n = 23.5 quarters, or about 6 years
So, 5.9 to be exact?
To determine how long it will take $1400 to double at 12% annual interest compounded quarterly, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Final amount (double the initial amount)
P = Principal amount (starting amount)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
In this case, P = $1400, r = 0.12 (12% expressed as a decimal), n = 4 (quarterly compounding), and we want to find t.
To double the initial amount, the final amount (A) will be $2800.
Substituting the values into the formula:
2800 = 1400(1 + 0.12/4)^(4t)
Divide both sides of the equation by 1400:
2 = (1 + 0.12/4)^(4t)
Now we can use logarithms to solve for t. Taking the natural logarithm of both sides:
ln(2) = ln[(1 + 0.12/4)^(4t)]
By applying the logarithmic property, we can bring down the exponent:
ln(2) = 4t * ln(1 + 0.12/4)
Divide both sides of the equation by 4 * ln(1 + 0.12/4):
t = ln(2) / (4 * ln(1 + 0.12/4))
Using a calculator, the result is approximately t = 5.79 years.
Therefore, it will take approximately 5.79 years for $1400 to double at a 12% annual interest rate compounded quarterly.