Today, Mark invested $5,000 into an account that guarantees 7.50% p.a., compounded monthly and Madonna invested $5,000 into account guaranteeing 8.125% p.a., compounded quarterly.

How long will it take (in years) for the value of Madonna's investment to be three times as much as Mark's?

For mark:

interest rate
= 7.5% p.a.
= 7.5%/12 per period of one month
= .625%
Future value of $5000 in n years, FVK(n)
= 5000*(1.00625)12*n

For Madonna,
interest rate
= 8.125% p.a.
= 8.125%/4 per quarter
= 2.03125% per quarter
Future value of $5000 in n years, FVD(n)
= 5000*(1.0203125)4*n

We look for n where
FVD(n) = 3FVK(n)
5000*(1.0203125)4*n
= 3*5000*(1.00625)12*n
(1.0203125)4*n / (1.00625)12*n = 3

Take log on both sides
4*n*log(1.0203125)-12*n*log(1.00625) = 3
n = log(3)/(4log(1.0203125-12log(1.00625))
= 193.785 years

Check:
in 193.785 years,
Mark will have $9,801,861,882.04
Madonna will have $29,405,482,100.99

To determine the number of years it will take for the value of Madonna's investment to be three times as much as Mark's, we can set up equations to represent the growth of their investments.

Let's denote the number of years as "t".

For Mark's investment, the formula to calculate the future value (FV) compounded monthly is:
FV = P * (1 + r/n)^(n*t)

Where:
P = Principal amount (initial investment) = $5,000
r = Annual interest rate = 7.50% = 0.075 (expressed as a decimal)
n = Number of times the interest is compounded per year = 12 (monthly compounding)
t = Time in years

Therefore, the value of Mark's investment after t years would be:
FV_Mark = $5,000 * (1 + 0.075/12)^(12*t)

Similarly, for Madonna's investment, the formula to calculate the future value compounded quarterly is:
FV = P * (1 + r/n)^(n*t)

Where:
P = Principal amount (initial investment) = $5,000
r = Annual interest rate = 8.125% = 0.08125 (expressed as a decimal)
n = Number of times the interest is compounded per year = 4 (quarterly compounding)
t = Time in years

Therefore, the value of Madonna's investment after t years would be:
FV_Madonna = $5,000 * (1 + 0.08125/4)^(4*t)

Since we want to find the number of years needed for Madonna's investment to be three times as much as Mark's, we can set up the equation:

3 * FV_Mark = FV_Madonna

Substituting the formulas for FV_Mark and FV_Madonna, we have:
3 * $5,000 * (1 + 0.075/12)^(12*t) = $5,000 * (1 + 0.08125/4)^(4*t)

Now we can solve for t.