The U.S. senate has 100 members. After a certain election,there were 12 more democrats than republicans, with no other parties represented. How many members of

Question

The U.S. senate has 100 members. After a certain election,there were 12 more democrats than republicans, with no other parties represented. How many members of each party were there in the senate?

Answer 1

The key to solving word (story) problems is to correctly translate the words and phrases into precise, concise math expressions (with variables and operations).

 

Assign variables (the problem asks us to find the number in each party).  Let:

    D = number of democrats

    R = number of republicans

 

Translate:

  "U.S. senate has 100 members ...no other parties represented"     means

             D + R = 100          (eq1)

  "12 more democrats than republicans"     means

     12   =   D              -       R           (eq2)

(Note:  eq2 could also be written D=R+12 or  R=D-12;   be sure you understand it)

 

Now, the easy math.  Solve using either substitution or elimination.  Let's use elimination:

           D       +       R     =     100           (eq1)

           D       -        R     =       12           (rewritten eq2)

      -------------------------------------      (elimination; add equations)

           2D                     =       112

             D                    =          56         (divide both sides by 2)

             D      -       R    =          12           (rewritten eq2 again)

        -----------------------------------       (elimination; now subtract equations)

                             R     =          44

 

To check:    56 + 44 = 100   ?

                       100 =  100   ?  yes

 Does      56 - 44 = 12   ?

                   12  = 12   ?   yes