# All other ratios from just a small set

1. Dec 29, 2016

### caters

1. The problem statement, all variables and given/known data
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
2. Relevant 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

3. 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?

Last edited by a moderator: Dec 30, 2016
2. Dec 29, 2016

### RUber

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:
$b/d = b/a * a/d = 1/5* 1/5 = 1/25$

3. Dec 29, 2016

### Ray Vickson

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.

4. Dec 29, 2016

### Buffu

Mod note: I fixed the TeX in the first post.
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.

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

Last edited by a moderator: Dec 30, 2016
5. Dec 30, 2016

### haruspex

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.

6. Jan 25, 2017

Staff Emeritus
I ran into this while searching for something else:

This is already an inconsistent set of conditions. No solution.

7. Jan 25, 2017

### haruspex

Why? 1 cat, 5 dogs, 25 mice.

8. Jan 25, 2017

### Buffu

9. Jan 25, 2017

### haruspex

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.