To calculate how long it will take for the value of Madonna's investment to be three times as much as Mark's, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = final amount (3 times the initial investment)
P = principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
For Mark's investment:
P1 = $5,000
r1 = 7.50% p.a. = 0.075
n1 = 12 (compounded monthly)
For Madonna's investment:
P2 = $5,000
r2 = 8.125% p.a. = 0.08125
n2 = 4 (compounded quarterly)
We need to find the value of t when A2 is three times larger than A1.
Step 1: Set up the equation
3P1 = P2(1 + r2/n2)^(n2t)
Step 2: Divide both sides by P1
3 = (P2/P1)(1 + r2/n2)^(n2t)
Step 3: Substitute the values
3 = (5000/5000)(1 + 0.08125/4)^(4t)
Step 4: Simplify
3 = (1.00)(1.0203125)^(4t)
Step 5: Take the logarithm of both sides
log(3) = log(1.0203125)^(4t)
Step 6: Apply the power rule of logarithm
log(3) = (4t) * log(1.0203125)
Step 7: Divide both sides by 4 * log(1.0203125)
t = log(3) / (4 * log(1.0203125))
Using a calculator:
Step 8: Calculate t
t ≈ log(3) / (4 * log(1.0203125))
t ≈ 19.97 years
Therefore, it will take approximately 19.97 years for the value of Madonna's investment to be three times as much as Mark's.