A thirty-year annuity X has annual payments of $\$1,000$ at the beginning of each year for twelve years, then annual payments of $\$2,000$ at the beginning of each year for eighteen years. A perpetuity $Y$ has payments of $Q$ at the end of each year for twenty years, then payments of $3Q$ at the end of each year thereafter. The present values of $X$ and $Y$ are equal when calculated using an annual effective discount rate of $10\%$ . Find $Q$.
We have that the discount rate is $d=.1$. Therefore $i=\frac{.1}{.9}=.11$. Our present values are:
$$X=1000\ddot{a}_{\overline{12}|i}+2000\ddot{a}_{\overline{18}|i}(1+i)^{-12}$$$$Y=Qa_{\overline{20}|i}+3Qa_{\overline{\infty}|i}$$
My instructor then writes that if we set these equal to each other we get $12040.14=11.75Q$. I haven't the slightest on how he simplified $Y$ to $11.75Q$, I've tried all the formulas for present value of annuity and perpetuity but it's not correct.
The expression you wrote for the present value of the perpetuity is not correct, because the present value of the payments of $3Q$ should be discounted by the $20$ years of deferral that passes after the first $20$ annual payments of $Q$ are made. That is to say, $$Y = Qa_{\overline{20}\rceil i} + 3Q v^{20} a_{\overline{\infty}\rceil i},$$ where $v = (1+i)^{-1} = 1 - d = 0.9$ is the present value discount factor. Alternatively, because the interest rate is fixed across all payments, you may equivalently regard the cash flow $Y$ as comprising two level perpetuities-immediate, one that pays $Q$, and the other paying $2Q$ after $20$ years: $$Y = Qa_{\overline{\infty}\rceil i} + 2Q v^{20} a_{\overline{\infty}\rceil i} = Q(1+2v^{20})a_{\overline{\infty}\rceil i} = \frac{1+2v^{20}}{i} Q,$$ which is a simpler expression to evaluate. In both cases, the result is $11.1884Q$.