All other ratios from just a small set

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The discussion revolves around calculating pet ratios based on given prevalence data, particularly focusing on the dog-to-cat and dog-to-rodent ratios. A participant initially misapplies a formula to derive the cat-to-rodent ratio, concluding it to be 1:5, which contradicts the expected outcome of 1:25 based on the dog-to-cat ratio. The conversation highlights the inconsistency in the provided ratios, as they do not align logically with the stated prevalence of dogs as the most common pet. Participants suggest that the complexity of the equations is unnecessary, advocating for simpler transitive relations to derive the ratios. Ultimately, the discussion concludes that the set of ratios is inconsistent and does not yield a valid solution.
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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.