Albert is saving up money for a down payment on a house. He currently has $4718, but knows he can get a loan at a lower interest rate if he can put down $5323. If he invests the $4718 in an account that earns 5% annually, compounded quarterly, how long will it take Albert to accumulate the $5323? Round your answer to 2 decimal places.
See previous post: Fri, 10-9-15, 2:o4 PM.
To find out how long it will take Albert to accumulate the desired amount of $5323, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (desired amount)
P = the initial principal (current amount)
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
In this case, Albert's initial principal is $4718, the annual interest rate is 5% (or 0.05 in decimal form), the interest is compounded quarterly (n = 4), and the desired amount is $5323.
Using the formula, we can solve for t:
5323 = 4718(1 + 0.05/4)^(4t)
Divide both sides by 4718:
5323/4718 = (1 + 0.05/4)^(4t)
Now take the natural logarithm (ln) of both sides to isolate the exponent:
ln(5323/4718) = ln[(1 + 0.05/4)^(4t)]
Using properties of logarithms, we can bring the exponent down:
ln(5323/4718) = 4t * ln(1 + 0.05/4)
Now divide both sides by 4 * ln(1 + 0.05/4):
t = ln(5323/4718) / (4 * ln(1 + 0.05/4))
Calculating this on a calculator or computer:
t ≈ 2.08
Therefore, it will take approximately 2.08 years for Albert to accumulate the desired amount of $5323.