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?
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.