On January 1, 2011, Tina invested P into a fund which accumulates at an interest rate of 3% compounded every 2 months. On January 1, 2012, Sam invested 50 in a fund with a discount rate of 6% compounded semiannually. On January 1,2011, the sum of the present value of the two funds is 100. Calculate the total combined amount that will be in the two funds on January 1, 2014. A)106 B)108 C)110 D)112 E)114
So besides the actual arithmetic, the format of this question itself is not very straight forward for me. I'm not understanding how Sam can have money in his account in 2011 at all if he didn't deposit anything until 2012. Any ideas?
As noted in the comment, when computing the present value of a cash flow, you are not computing an actual balance in the sense of a realized amount of money. Instead, you are computing the equivalent time value of money at a specified point in time.
To this end, we note that if $P$ is the present value of Tina's fund on 01 Jan 2011, and $Q$ is the present value of Sam's fund on that same date, then $P+Q = 100$. Since the value of Sam's fund is $50$ on 01 Jan 2012, where the nominal semiannual discount rate is $d^{(2)} = 0.06$, then the effective annual rate of discount is $$d = 1- (1 - d^{(2)}/2)^2 = 0.0591.$$ This means that $$50(1-d) = Q,$$ or $Q = 47.045$. Equivalently, we can observe that the effective annual interest rate is $$i = \frac{d}{1-d} = 0.0628122,$$ thus $Q(1+i) = 50$ and the answer is the same. Consequently, $P = 100-Q = 52.955$, and the accumulated value on 01 Jan 2014 of both funds is $$P\left(1 + \tfrac{i^{(6)}}{6}\right)^{18} + Q(1+i)^3 = (52.955)(1.005)^{18} + (47.045)(1.0628122)^3 \\= 57.929 + 56.4785 = 114.4.$$ Note we could have obtained the accumulated value of Sam's fund by taking the value at initial deposit on 01 Jan 2012 and discounting it by two years; i.e., its accumulated value is also $50/(1-d)^2 = 50/(1-0.0591)^2 = 56.4785$.