All other ratios from just a small set

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SUMMARY

The discussion centers on calculating pet ratios based on given relationships among dogs, cats, rodents, and other pets. The initial ratios provided include Dog:Cat = 5:1 and Dog:Rodent = 1:5. Participants identified inconsistencies in the calculations, particularly regarding the Cat:Rodent ratio, which should be 1:25 instead of 1:5. The conversation highlights the importance of correctly applying ratio transitivity and recognizing the prevalence of each pet type in the calculations.

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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.