# All other ratios from just a small set

• caters
No solution.In summary, the person noticed that dogs were the most prevalent pet in a world full of pets. However, he also observed 8 ratios, which included the ratio of dogs to cats, dogs to rodents, dogs to lizards, dogs to snakes, dogs to turtles, dogs to alligators, dogs to crocodiles, and dogs to birds. These ratios were inconsistent and did not have a solution.
caters

## 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.

Mod note: I fixed the TeX in the first post.
caters said:
Did you mean ?
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.

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.

## 1. What are ratios?

Ratios are mathematical expressions that compare the sizes of two quantities. They are typically written in the form of a fraction, with the first number representing the quantity being measured and the second number representing the reference quantity.

## 2. How are ratios different from fractions?

Ratios and fractions are similar in that they both compare two quantities. However, ratios are typically used to compare two different units of measurement, while fractions are used to represent parts of a whole.

## 3. Can ratios be simplified?

Yes, ratios can be simplified just like fractions. To simplify a ratio, you can divide both numbers by their greatest common factor (GCF). This will result in an equivalent, simplified ratio.

## 4. How are ratios used in real-life situations?

Ratios are used in a variety of real-life situations, such as in cooking, finance, and sports. For example, a recipe may call for a ratio of 2 cups of flour to 1 cup of sugar, and a financial statement may include a ratio of a company's assets to liabilities. In sports, ratios are often used to measure a player's performance, such as the ratio of goals scored to games played.

## 5. Are there different types of ratios?

Yes, there are different types of ratios, including unit ratios, equivalent ratios, and rates. Unit ratios compare two quantities with the same unit of measurement, equivalent ratios have the same value but may have different units, and rates compare quantities with different units, typically using the word "per" in between the numbers.