Rate of Return

Jill has been putting money into a mutual fund for years. Her income went up and down, so the amount she invested changed every time, and she did not invest on a fixed schedule. She does have a complete record of when she invested and how much.

The fund sends her a report now and then with the current value of the account. Jill wonders whether the same money would have done better in a savings account paying a fixed rate. To decide that she first needs the fixed rate that would have produced the same value. Compute that rate.

Assume Jill added money only at the beginning of a month and that every month has the same length. Interest on a savings account is paid at the end of each month and compounds monthly, so with a monthly rate of $i$, an amount of $a$ dollars added at the beginning of month $m$ is worth $a(1+i)^{M-m+1}$ at the end of month $M$.

Jill invested $N$ times, adding $a_k$ dollars at the beginning of month $m_k$, and at the end of month $M$ the fund is worth $V$ dollars. Find the monthly rate $i$ that satisfies

$\sum_{k=1}^{N} a_k (1+i)^{M - m_k + 1} = V$

For instance, if Jill adds $100 at the beginning of month 1 and another $100 at the beginning of month 3, and the fund is worth $210 at the end of month 4, the equation becomes $100(1+i)^4 + 100(1+i)^2 = 210$, whose root is about $0.016351795234$.

The input contains several cases. Each case begins with an integer $N$ ($1 \le N \le 12$), the number of times Jill added money. After it come $N + 1$ pairs, each pair an integer followed by a real number. The integer is a month number (1 or larger) and the real number is a dollar amount. The first $N$ pairs give the month and the amount of each investment, in increasing order of month number. The last pair gives the month $M$ of the report and the value $V$ of the fund at the end of that month.

One or more whitespace characters (spaces, tabs, line breaks) separate the numbers, so a pair may be split across two lines.

A single integer $-1$ follows the last case.

For each case print one line in the form Case k: r, where $k$ is the case number counting from 1 and $r$ is the equivalent fixed monthly rate rounded to five digits after the decimal point. Print one blank line between consecutive cases.

The rate is at least 0 and at most 1. No case has an exact rate that lands on a rounding boundary, so the five printed digits are determined uniquely.