Question

Tasha invests $5,000 at 6% annual interest and an additional $5,000 at 8% annual interest. Thomas invests $10,000 at 7% annual interest. Which statement accurately compares Tasha’s and Thomas’s investments if interest is compounded annually?

Compound interest formula: V (t) = P (1 + StartFraction r Over n EndFraction) Superscript n t
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
Each person will have exactly the same amount over time because each invested $10,000 at an average interest rate of 7%.
Tasha’s investment will yield more over many years because the amount invested at 8% causes the overall total to increase faster.
Thomas’s investment will yield more from the start because he has more money invested at the average percentage rate.
Tasha’s investment will yield more at first because she invested some at a higher rate, but Thomas’s investment will yield more over the long run.

Answer 1

Answer: Tasha’s investment will yield more over many years (choice 2).

Reason: After t years
Tasha: V_Tasha = 5000(1.06)^t + 5000(1.08)^t
Thomas: V_Thomas = 10000(1.07)^t

At t = 0 and t = 1 the amounts are equal. For t > 1 the function a -> a^t is convex, so by Jensen's inequality
(1.06)^t + (1.08)^t > 2(1.07)^t,
hence V_Tasha > V_Thomas for all t > 1. (For 0 < t < 1 the inequality reverses, so Thomas would be slightly larger for fractional years less than 1.)