# Dumb Question on ratio

## Main Question or Discussion Point

Dumb Question on "ratio"

I have some text that implies the following (which makes abolutely no sense to me).

The ratio of 1/12, 5/12, 7/12, 11/12 is 4/11

Can anyone think of any context or meaing of "ratio" here for which this statement would make any sense?

berkeman
Mentor

Weird. The increment for that sequence is almost 4/12, but not in one case.

Can you tell us what the context of the statement is? Where did it come from?

Hi berkeman.

The text of the problem starts out as follows :

We shall call a fraction that cannot be cancelled down a resilient fraction.
Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of its proper fractions that are resilient; for example, R(12) = 4⁄11.
When I read this my understanding is that the proper fractions of denominator 12 which are "resilient" would be 1/12, 5/12, 7/12 and 11/12. But then how could one define the ratio of those to be 4/11.

In other words this is my problem:
- I think I understand how the author is defining "resilient" fractions.
- I think I understand what the author calls "the proper fractions of a denominator that are resilient".
- But I still don't understand how R(d) is defined or how that example works.

Perhaps it's just badly worded and I am totally misunderstanding the whole thing.

Last edited:
tiny-tim
Homework Helper
Hi uart!
We shall call a fraction that cannot be cancelled down a resilient fraction.
Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of its proper fractions that are resilient; for example, R(12) = 4⁄11.
Right … 12 has 11 proper fractions: 1/12, 2/12, … 11/12.

And 4 of them are resilient … 1/12, 5/12, 7/12, 11/12.

So the ratio is 4/11.

berkeman
Mentor

Oh, I think I see. There are 4 numerators out of the possible 11 numerators of x/12 that are resiliant. Weird way of defining things. I wonder if it's useful somehow later...?

Edit -- TT beats me to the punch again!!

Thanks to both :).

So I guess the text could have been better worded as:

"Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of the number of its proper fractions that are resilient to the total number of it's proper fractions; for example, R(12) = 4⁄11."

I think this question arose from problem 243 of Project Euler (projecteuler.net)

HallsofIvy
Homework Helper

Or "R(d) is the number of integers less than d that are relatively prime to d".

The whole statement, as given, sounds like something made up by a school boy.

Added: Ah, yes, I checked "project Euler" and that is precisely what it is.