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

• uart
In summary, the conversation is discussing the definition of "resilient fractions" and the ratio of "resilient" proper fractions to the total number of proper fractions. This concept is used in problem 243 of Project Euler and was initially confusing but was clarified by Tiny-Tim's explanation.
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?

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 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.

Last edited:
Hi uart!
uart said:
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.

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)

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.

perfectno28 said:
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.

## 1. What is a ratio?

A ratio is a comparison between two quantities, usually expressed as a fraction. It tells us how many times one value is contained within another value.

## 2. How do you calculate a ratio?

To calculate a ratio, you divide one quantity by another. For example, if there are 4 red apples and 8 green apples, the ratio of red apples to green apples would be 4/8 or 1/2.

## 3. What is the difference between a ratio and a proportion?

A ratio is a comparison between two values, while a proportion is an equation that states two ratios are equal. In other words, a proportion is a statement that two ratios are equivalent.

## 4. How can ratios be used in real life?

Ratios are used in many real-life situations, such as measuring ingredients in a recipe, calculating distances on a map, or determining the price of an item per unit. They are also commonly used in financial analysis and in scientific experiments.

## 5. Can a ratio be simplified?

Yes, ratios can be simplified just like fractions. To simplify a ratio, you divide both the numerator and denominator by their greatest common factor (GCF).

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