# Calculating Perpetuities

Sandy purchases a perpetuity-immediate that makes annual payments. The first payment is 100, and each payment thereafter increases by 10. Danny purchases a perpetuity-due which makes annual payments of 180. Using the same effective interest rate, i>0, the present value of both perpetuities are equal. Calculate i.

This is from the Study Manual for Exam FM/Exam 2 Eleventh Edition Section 4h and 4i number 3. This whole section has been very confusing for me and I don't quite understand the reasoning. The provided answer provided is 10.2%. If anyone could help me out I would really appreciate it!

$$\underbrace{\frac{100}{i}+\frac{10}{i^2}}_{\textrm{Sandy}}=\underbrace{\frac{180}{d}}_{\textrm{Danny}}$$ where $d=\frac{i}{1+i}$. So we have to solve $$180i^2 + 80i – 10 = 0\quad\Longrightarrow\quad 18i^2+8i-1=0$$ and we find $$i=\frac{-4\pm \sqrt{4^2-18\times(-1)}}{18}=\begin{cases}\frac{-4+\sqrt{34}}{18}=10.17\%& \Longrightarrow\quad\text{OK}\\ \frac{-4-\sqrt{34}}{18}<0 & \Longrightarrow\quad\text{discard}\end{cases}$$