Question

Jacques deposited $1,900 into an account that earns 4% interest compounded semiannually. After t years, Jacques has $3,875.79 in the account. Assuming he made no additional deposits or withdrawals, how long was the money in the account?

Compound interest formula:mc007-1.jpg

t = years since initial deposit
n = number of times compounded per year
r = annual interest rate (as a decimal)
P = initial (principal) investment
V(t) = value of investment after t years
2 years
9 years
18 years
36 years

Answer 1

Use A = P(1 + r/n)^{nt}. Solve for t:

t = ln(A/P) / (n ln(1 + r/n))

Here P = 1900, A = 3875.79, r = 0.04, n = 2, so 1 + r/n = 1.02.

t = ln(3875.79/1900) / (2 ln 1.02)
≈ ln(2.03988474) / (2 · 0.01980263)
≈ 0.7128946 / 0.03960525
≈ 18

So the money was in the account for 18 years.