# Period of a Fraction

From what I have seen, the longest length of a repeating sequence for an irrational number is c-1 for a=b/c. This occurs when c is a prime. How does one prove this? Can you give mathematical proof for this?

Here is a link to the problem being discussed: http://boards.straightdope.com/sdmb/showthread.php?t=720360

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#### Solution Preview

For the fixed a = b/c, let's define a function f(n) as follows. We take the digits of the decimal expansion after the first n digits behind the decimal point and move them immediately after the decimal point. The number we then get is then f(n). So if b/c = 231.03245324532956...., then f(1)=0.3245324532956...., f(2)= 0.245324532956...., f(3) = 0.45324532956.... We define f(0) as the fractional part. The question is then how long the period of the function f(n) can be. Now f(n) can be expressed as the fractional part of (b/c)*10^n, which we can write as:

f(n) = (b*10^n mod c)/c

Here A mod B (pronounced as A modulo B) stands for the remainder of A after division by B. E.g. 20 mod 7 = 6. Now, when we do computations modulo some fixed number c, it's useful to work with only the numbers that can have remainders after the division by c and define addition and multiplication on those numbers, such that this will be consistent with ordinary addition and multiplication, and then take the remainder. One way to quickly arrive at the desired formalism is to consider ...

#### Solution Summary

This response gives detailed non-technical proof for the maximum length of the period of a fraction.

Caledonia: calculate payback period, NPV, IRR, and ranking conflict

I do not understand the NPV, or IRR. I have to answer these questions in an excel spreadsheet.

File is attached.

12. Caledonia is considering two additional mutually exclusive projects. The cash flows associated with these projects are as follows:

YEAR PROJECT A PROJECT B

0 −$100,000 −$100,000

1 32,000 0

2 32,000 0

3 32,000 0

4 32,000 0

5 32,000 $200,000

The required rate of return on these projects is 11 percent.

a. What is each project's payback period?

b. What is each project's net present value?

c. What is each project's internal rate of return?

d. What has caused the ranking conflict?

e. Which project should be accepted? Why?