Understanding the Confusing Concept of Ratio: A Beginner's Guide

[uart]

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]

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?

 

[uart]

Hi berkeman.

The text of the problem starts out as follows :

We shall call a fraction that cannot be canceled 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.

 

[tiny-tim]

Hi uart!

We shall call a fraction that cannot be canceled 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]

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!

 

[uart]

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."

 

[perfectno28]

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

 

[HallsofIvy]

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.

 

[uart]

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

Yes someone asked me about that particular Project Euler problem and I had trouble making sense of their wording. Tiny-Tim's answer above made it clear though.