Division for Ratios: Explaining Value of Fractions

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Discussion Overview

The discussion revolves around the use of division in understanding ratios, particularly in the context of comparing quantities, such as dogs to cats. Participants explore the significance of expressing ratios as fractions and the implications of different numerical representations of these ratios.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions why division is used for ratios and seeks to understand the relationship between the fraction representation (1/2) and the ratio of 1 dog to 2 cats.
  • Another participant suggests that ratios provide an easier understanding of relationships between quantities, especially when dealing with larger numbers, and notes that ratios can also represent percentages.
  • A different participant reiterates that the ratio of 1 dog for every 2 cats can be expressed as 1/2 a dog for every 1 cat, emphasizing the significance of both 1/2 and 2 in this context.
  • One participant challenges the need for division by asking if a verbal explanation would be preferable, implying a preference for clarity in communication.
  • Another participant states that a "ratio" is defined as a fraction, which implies division.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and clarity of using division for ratios, with some supporting its use while others question it. The discussion remains unresolved regarding the best approach to communicate ratios effectively.

Contextual Notes

Participants have not fully explored the implications of using different numerical representations of ratios, and there is a lack of consensus on the clarity and effectiveness of division in this context.

Mr Davis 97
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Why is division used for ratios? For example, if we have 1 dog for every 2 cats, then why can this relation be modeled by the division? Since the ratio can essentially have the value of 1/2 or 2, then what is the significance of these two distinct numbers in relation to the ratio? How does the value of the fraction 1/2, which is 0.50, relate to the ratio 1 dog to 2 cats? If we said we had a 0.50 ratio of dogs to cats, what does that mean?
 
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There are different reasons. First of all the ratio is a much easier number to understand, depending if it is greater or smaller than 1 (and how much grater/smaller it is) you can have an immediate idea of what's going on. If you are using small numbers (as in your example) it is clear that the ratio doesn't really add any information. However if the numbers you are dealing with are high it is much more complicated to evaluate the relationship between the two quantities just "by eye".

Moreover, the ratio gives you the percentage. For example, as you said, if I know that the ratio dogs/cats is 0.37 then this means that the number of dogs is the 37% of the number of cats.
 
Mr Davis 97 said:
Why is division used for ratios?
Because it works.

Mr Davis 97 said:
For example, if we have 1 dog for every 2 cats, then why can this relation be modeled by the division?
Because 1 dog for every 2 cats is the same as ## \frac12 ## a dog for every 1 cat.

Mr Davis 97 said:
Since the ratio can essentially have the value of 1/2 or 2, then what is the significance of these two distinct numbers in relation to the ratio?
The significance of ## \frac12 ## is that we have ## \frac12 ## a dog for every 1 cat, and the siginficance of 2 is that we have 2 cats for every 1 dog.

Mr Davis 97 said:
If we said we had a 0.50 ratio of dogs to cats, what does that mean?
That we have 0.5 dogs for every cat (which is the same as 1 dog for every 2 cats, or 2 cats for every 1 dog etc...)
 
Mr Davis 97 said:
Why is division used for ratios?

What would you prefer to use? Do you want everyone to spell it out in words every time?
 
A "ratio" is defined as a fraction which is an implied division.
 

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