An animal shelter has a total of 120 animals.

Question

An animal shelter has a total of 120 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter?

Answer 1

This one is slightly more work because it is a linear system in three variables. Normally, the strategy for solving a linear system is to combine equations through substitution or addition to reduce both the number of variables and equations by one. You do it again until you are down to a 1 x 1. Solve, then climb back up the stairs you just climbed down.

Let's organize:

Cats: C

Dogs: D

Rabbits: R

C + D + R = 120

R = C/2 - 5   (because we said "than", which makes us write in reverse)

C = D + 20

It turns out this one is insultingly easy, and hardly worth the ticket price. Merely plug the cat equation into the rabbits:

R = (D + 20) / 2 - 5   = D/2 + 10 - 5  = D/2 + 5

Plugging everything into the main equation gives us:

D + 20 + D + D/2 + 5 = 120

2.5D + 25 = 120                   Isolate the D, starting with 25..

        -25       -25

2.5D         = 95                    ...then divide...

2.5               2.5

    D        = 38

That means cats are:

C = 38 + 20 = 58

and rabbits...

R = 58/2 - 5 = 29 - 5 = 24

Double-checking, we have:

58 + 38 + 24 = 120

        120       = 120