How many dogs are on the farm?

Question

There are 38 animals on a farm, all chickens and dogs. There are 92 legs in all. Show work please.

Answer 1

Algebra provides a way to put complicated descriptions into equations using variables and operations (like multiply, add, etc.).  To “solve” a system of equations means to find the value(s) of the variables that make all the equations true at the same time.

The units of the equations are important:

      ( (2 legs)/(1 chicken) ) * (number of chickens) = number of legs for chickens

      ( ( 4 legs)/(1 doc) ) * (number of dogs) = number of dog legs

 

If we made a table, we could be here for a long time:

      Number of chickens     Number of dogs     Total number of legs

       -------------------------     ---------------------       ---------------------------

                   0                                   38                                   152

                   1                                   37                                   150

                   2                                   36                                   148

                   3                                   35                                   146

        . . .

                  30                                    8                                     92

        . . .

                  38                                    0                                     76

 

Let:

  C= number of chickens

  D = number of dogs

 

Translate:

  “38 animals on a farm, all chickens and dogs”      means    C + D = 38      [eq1]

  “There are 92 legs in all”    means

                    (number of chicken legs) + (number of dog legs) = 92

                            2C                             +           4D              =        92     [eq2]

  “How many dogs are on the farm?”     means      report D

 

The math:

  We may either substitute or eliminate to solve for one variable.  Let’s eliminate C and solve for D (which is what the question asks us to find):

          2C    +   2D   =   76               [2*eq1]

          2C    +   4D   =   92                [eq2 again]

       -----------------------------     [elimination; subtract equations]

                      -2D   =   -16

                          D =  8         [divide both sides by (-2)]

    There are 8 dogs on the farm.

                               [note that this agrees with the table above]

 

The problem doesn’t asks for C, but to check our answer:

    Is   30 + 8 = 38   ?

             38  =   38   ?yes

    Is   2(30) + 4(8) = 92   ?

           60  +  32   =  92   ?

                92  = 92   ?yes