Perpetuity Immediate Present Value Question 
A perpetuity-immediate pays $X per year.  Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments.  Brian's share of the present value of the original perpetuity is 40%, and Jeff's share is K.  Calculate K.

The perpetuity immediate present value is:  
$a_{\infty\neg i}=\cfrac{X}{i}\tag{1}$
$\text{Brian's share}=0.4\cfrac{X}{i}\tag{2}$
I don't know where to go from here.  Can anyone please help or explain?  Thank you.
 A: Hint 1: You are correct about Brian's share, but there is another expression that can give the value of Brian's payments, which is the present value of an $n$ year annuity immediate that pays $X$ per year.  Set this equal to the $0.4X/i$.  You will need this equation after Hint 2.
Hint 2: Jeff gets every payment after the first $2n$ payments.  So, he gets a perpetuity that starts $2n$ years later, with payments of $X$.  The present value of this is then
$$\frac{X}{i} v^{2n}.$$  You need to figure out this as a percent of $\frac{X}{i}$, so in other words you need to know the value of $v^{2n}$.  Use your equation from Hint 1.
A: Split the entire perpetuity-immediate into three different segments. Brian gets an annuity-immediate with $n$ payments, so Brian's present value is going to be
$$Xa_{\overline{n}|} = 0.4\left(\dfrac{X}{i}\right)\text{.}$$
Colleen receives the second stream of $n$ payments, the last of which is paid at the end of $2n$ years from now. So we do the same as Brian, except we multiply by $v^{n}$ since $a_{\overline{n}|}$ on its own will value her payments at time $n$. Furthermore, we know that however much her share of the perpetuity is, it will satisfy the equation
$$Xa_{\overline{n}|}v^{n} = \left(1-0.4-K\right)\left(\dfrac{X}{i}\right)\text{.}$$
Jeff's share is a perpetuity with the first payment occurring at time $2n+1$ so that using the perpetuity-immediate formula will time at time $2n$. To value it at time 0, we have:
$$\left(\dfrac{X}{i}\right)v^{2n} = K\left(\dfrac{X}{i}\right)\text{.}$$
You have everything you need here - just remember that $a_{\overline{n}|} = \dfrac{1-v^n}{i}$.
A: This might just be happenstance, but I came across this problem, too, and noticed that $(1 - 0.4) \cdot 0.4 = 0.24$ gives Colleen's share, making Jeff's share equal to $1 - 0.24 - 0.4 = 0.36$. I think it makes sense to think that the first $n$ payments constitute $40\% $ of the present value. So if $1 - 0.4 = 0.6$ of the present value is left after Brian's share, then Colleen's share of what's left, as another $n$ payments, would just be $0.4 * 0.6 = 0.24$ of the present value, making Jeff's share equal to $0.36$. I dunno. Made sense in my head.
