Suppose that you invested $1,442 at the annual rate of 4.60% compounded continuously, and your friend invested $885 at the annual rate of 6.35% compounded quarterly. In how quarters will your friends investment exceeds yours?
To determine in how many quarters your friend's investment will exceed yours, we need to compare the growth of both investments over time.
First, let's find the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the amount after time t
P = the principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = the annual interest rate (expressed as a decimal)
t = time (in years)
For your investment:
P1 = $1,442
r1 = 4.60% = 0.046 (as a decimal)
For your friend's investment:
P2 = $885
r2 = 6.35% = 0.0635 (as a decimal)
Now, at any given time t, your investment is:
A1 = P1 * e^(r1t)
For your friend's investment compounded quarterly, we'll use the formula for compound interest:
A = P * (1 + r/n)^(nt)
Where:
A = the amount after time t
P = the principal amount (initial investment)
r = the annual interest rate (expressed as a decimal)
n = the number of times compounded per year
t = time (in years)
For your friend's investment:
A2 = P2 * (1 + r2/4)^(4t) [compounded quarterly]
We want to find the number of quarters (n) when your friend's investment exceeds yours, so we'll set up and solve the inequality:
A2 > A1
P2 * (1 + r2/4)^(4t) > P1 * e^(r1t)
Now, let's substitute the given values into the inequality:
$885 * (1 + 0.0635/4)^(4t) > $1,442 * e^(0.046t)
Now we can solve this inequality for t.