All other ratios from just a small set

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Homework Help Overview

The problem involves determining additional ratios based on a set of given ratios related to the prevalence of various pets, specifically dogs, cats, rodents, lizards, snakes, turtles, alligators, crocodiles, and birds. The context is centered around understanding the relationships between these animals through their ratios.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the calculation of the Cat:Rodent ratio and question the validity of the original poster's formula. Some suggest that the Cat:Rodent ratio should be 1:25 instead of 1:5, raising concerns about the interpretation of the ratios. Others point out inconsistencies in the provided ratios, questioning the overall setup.

Discussion Status

The discussion is ongoing, with participants examining the original equations and ratios. Some have provided insights into potential errors in the calculations, while others have highlighted inconsistencies in the ratios that may lead to no viable solution. There is a mix of interpretations being explored without a clear consensus.

Contextual Notes

Participants note that the ratios presented may lead to contradictions regarding the prevalence of pets, particularly with respect to the number of dogs compared to other animals. This raises questions about the assumptions underlying the ratios.

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Homework Statement


A person has come to a world full of pets. He notices right away that dogs are the most prevalent of all pets. But that's not all. He notices these 8 ratios:

Dog:Cat = 5:1
Dog:Rodent = 1:5
Dog:Lizard = 1:3
Dog:Snake = 2:1
Dog:Turtle = 1:3
Dog:Alligator = 3:1
Dog:Crocodile = 7:1
Dog:Bird = 1:10

What are the other ratios based on these 8 ratios

Homework Equations



$$\frac{\frac{1}{a:b}}{a*\frac{1}{c:d}} = b:ad$$

where

a = antecedent of first ratio
b = consequent of first ratio
c = antecedent of second ratio
and
d = consequent of second ratio

The Attempt at a Solution


Let's do the Cat:Rodent ratio first

$$\frac{\frac1 5}{5*\frac 5 1}$$

Thus the Cat:Rodent ratio is 5:25 or 1:5

But this doesn't seem right. If cats are 5 times less prevalent than dogs, how can the Cat:Rodent ratio equal the Dog:Rodent ratio? Surely the Cat:Rodent ratio should be 1:25

So what is wrong with my ratio calculating formula that is based on 2 known ratios?
 
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I don't follow your source equation. If a is the antecendent in the first ratio, and b is the consequent of the first ratio, aren't you looking to find b:d? Why do you write b:ad?
So you have a:b and a:d, and you want b:d. There is no need for a c in the equation.
Your intuition is correct, it should be 1:25, this would be summarized by b:d where:
## b/d = b/a * a/d = 1/5* 1/5 = 1/25##
 
caters said:

Homework Statement


A person has come to a world full of pets. He notices right away that dogs are the most prevalent of all pets. But that's not all. He notices these 8 ratios:

Dog:Cat = 5:1
Dog:Rodent = 1:5
Dog:Lizard = 1:3
Dog:Snake = 2:1
Dog:Turtle = 1:3
Dog:Alligator = 3:1
Dog:Crocodile = 7:1
Dog:Bird = 1:10

What are the other ratios based on these 8 ratios

Homework Equations



$$\frac{\frac{1}{a:b}}{a*\frac{1]{c:d}} = b:ad$$

where

a = antecedent of first ratio
b = consequent of first ratio
c = antecedent of second ratio
and
d = consequent of second ratio

The Attempt at a Solution


Let's do the Cat:Rodent ratio first

$$\frac{\frac{1}{5}}{5*\frac{5}{1}$$

Thus the Cat:Rodent ratio is 5:25 or 1:5

But this doesn't seem right. If cats are 5 times less prevalent than dogs, how can the Cat:Rodent ratio equal the Dog:Rodent ratio? Surely the Cat:Rodent ratio should be 1:25

So what is wrong with my ratio calculating formula that is based on 2 known ratios?

Let ##D, C, R, L, S, T, A , Cr, B## be the number of dogs, cags, rodents, lizards, snakes, turtles, alligators, crocodiles and birds in the world. We are told that
$$C = \frac{1}{5} D, \:R = 5 D, \ldots, \:B = 10 D,$$
so all other ratios like ##C:S = C/S##, etc., can be determined easily.
 
Your Latex is incorrect :(.
Mod note: I fixed the TeX in the first post.
caters said:
\frac{\frac{1}{a:b}}{a*\frac{1]{c:d}} = b:ad
Did you mean ?
##\displaystyle \frac{\frac{1}{a:b}}{a*\frac{1}{c:d}} = b:ad##
In which case ,
LHS ##\displaystyle = \frac{\frac{1}{\frac ab}}{a*\frac{1}{\frac cd}} = \frac{\frac{b}{a}}{\frac{ad}{c}} = \frac{b}{a} \times {\frac{c}{ad}} = \frac{bc}{a^2d} \ne {b\over ad} \ne## RHS.

caters said:
\frac{\frac{1}{5}}{5*\frac{5}{1}
##\frac{\frac{1}{5}}{5*\frac{5}{1}}##
 
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caters said:

Homework Equations



$$\frac{\frac{1}{a:b}}{a*\frac{1}{c:d}} = b:ad$$
that clearly cannot be right since you have a c on the left but no c on the right.
The simplest change to make it maybe valid is
$$\frac{\frac{1}{c:b}}{a*\frac{1}{c:d}} = b:ad$$
but I still don't like it because it assumes a ratio x:y is to be interpreted as the fraction x/y, not y/x.
Besides, as RUber points out, an equation connecting two unrelated ratios via a third is unnecessarily complex. You just need the transitivity relation a:b * b:c = a:c.
 
I ran into this while searching for something else:

caters said:
He notices right away that dogs are the most prevalent of all pets. But that's not all. He notices these 8 ratios:

Dog:Cat = 5:1
Dog:Rodent = 1:5

This is already an inconsistent set of conditions. No solution.
 
Vanadium 50 said:
I ran into this while searching for something else:
This is already an inconsistent set of conditions. No solution.
Why? 1 cat, 5 dogs, 25 mice.
 
haruspex said:
Why? 1 cat, 5 dogs, 25 mice.
caters said:
He notices right away that dogs are the most prevalent of all pets
 
Thanks, I somehow skipped that in reading Va's post. ( The dog:cat ratio was irrelevant.)
The turtle, lizard and bird ratios also violate the prevalence.