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.

4718(1+.05/4)^(4n) = 5323

(1.0125)^(4n) = 1.1282
4n = log(1.1282)/log(1.0125)
n = 2.4
or, roughly 2 1/2 years

math

the answer is 4

Well, Albert is certainly in a conundrum! He needs to reach a magical number of $5323, but he's currently stuck at $4718. Let's see how long he'll have to wait for his dream house.

To calculate the time it takes for Albert's investment to grow, we need to use the magic of compound interest. With an interest rate of 5% annually, compounded quarterly, the formula we need is:

A = P(1 + r/n)^(nt)

Where:
A = the final amount (5323 in this case)
P = the initial amount (4718)
r = the annual interest rate (5%)
n = the number of times the interest is compounded per year (quarterly, so 4 times)
t = the number of years

Now, let's plug in the numbers and find out how long Albert will have to wait:

5323 = 4718(1 + 0.05/4)^(4t)

Now, let's do some math magic and solve for t:

1.1275^(4t) = 5323/4718

4t = log(5323/4718) / log(1.1275)

t = (log(5323/4718) / log(1.1275)) / 4

Calculating this monstrous equation, we find that t ≈ 1.09 years.

So, it'll take Albert approximately 1.09 years to accumulate enough money for the down payment on his dream house. That's less than two years of saving! Good luck to Albert on his financial endeavor!

To find out how long it will take Albert to accumulate the desired amount, 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, $5323)
P = the principal amount (in this case, $4718)
r = the annual interest rate (5% or 0.05)
n = the number of times interest is compounded per year (quarterly, so 4 times a year)
t = the number of years

Rearranging the formula to solve for t:

t = log(A/P) / (n * log(1 + r/n))

Plugging in the values:

t = log(5323/4718) / (4 * log(1 + 0.05/4))

Using a calculator, we can get the value of t as follows:

t ≈ log(1.1294) / (4 * log(1.0125))

After evaluating the logarithms:

t ≈ log(1.1294) / (4 * log(1.0125))
≈ 0.0523 / (4 * 0.0031)
≈ 0.0523 / 0.0124
≈ 4.2149

Rounding the answer to 2 decimal places, it will take Albert approximately 4.21 years to accumulate the required amount of $5323.