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All other ratios from just a small set

  1. Dec 29, 2016 #1
    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. jcsd
  3. Dec 29, 2016 #2

    RUber

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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##
     
  4. Dec 29, 2016 #3

    Ray Vickson

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    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.
     
  5. Dec 29, 2016 #4
    Your Latex is incorrect :(.
    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
  6. Dec 30, 2016 #5

    haruspex

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    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.
     
  7. Jan 25, 2017 #6

    Vanadium 50

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    I ran into this while searching for something else:

    This is already an inconsistent set of conditions. No solution.
     
  8. Jan 25, 2017 #7

    haruspex

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    Why? 1 cat, 5 dogs, 25 mice.
     
  9. Jan 25, 2017 #8
     
  10. Jan 25, 2017 #9

    haruspex

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    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.
     
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