Question

Betty and Bob invest in separate accounts at the same time. Unfortunately Bob’s account decreases in value at a continuous rate per annum of magnitude 3%. Fortunately Betty’s account increases in value at a rate of 16% per annum compounded quarterly. If Betty initially invests 35% less than Bob then algebraically find how much time it will take for their accounts to be of equal value.
Your answer should be stated in years and be accurate to 2 places after the decimal point.

Answer 1

First of all we will calculate the annual effective rate of BOB;s investment

=1.04^4 = 16.9859%

Let amount invested by Betty = 100000

Therefore amount invested by BOB = 100000 - 35% of 100000 = 65000

let the time taken = t

future value of Berry's investment after n years = 100000 * (1-0.03)^t

Future Value of bob's investment = 65000*(1.169859)^t

Therefore we have the following equation

100000 * 0.97^t = 65000 * 1.169859^t

100000 / 65000 = 1.169859^t / 0.97^t

1.5384 = 1.2060^t

t = ln(1.5384) / ln(1.2060)

t = 0.4307 / 0.1873 = 2.299years

PROOF

FV of Betty's investment = 100000 * (1-0.03) ^ 2.299 = 93235.73

FV of Bob's investment = 65000 * (1.169859)^2.299 = 93235.65

Hence proved