word problem

Question

divide $ 106 among Shelly, Veronica, Christy and Ruth so that Shelly and Veronica together have $ 56, Shelly and Christy together have $ 59 and Shelly and Ruth together have $ 43.

Answer 1

Let:

    S = Shelly's amount

    V = Veronica's amount

    C = Christy's amount

    R = Ruth's amount

 

Translate:

  "divide $ 106 among Shelly, Veronica, Christy and Ruth"   

                 means  S + V + C + R = $106   [eq1]

  "Shelly and Veronica together have $ 56,"     means    S + V = $56    [eq2]

  "Shelly and Christy together have $ 59"      means   S + C = $59     [eq3]

  "Shelly and Ruth together have $ 43"       means  S + R = $43    [eq4]

 

The math:

  Add eq2, eq3, and eq4

        S  +  V                     =  $56

        S          +  C             =  $59

        S                   +  R    =  $43

  -----------------------------------

      3S  +  V  +  C  +  R   =  $158

        S  +  V  +  C  +  R   =  $106      [subtract eq1 to Eliminate V,C,R]

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

      2S                             =   $52

        S                             =   $26      [eq5]

 

      S + V = $56       [eq2]

      S       = $26        [eq5]

 --------------------     [Elimination; subtract equations]

            V = $30

 

       S + C = $59       [eq3]

       S        = $26       [eq5]

 --------------------     [Elimination; subtract equations]

              C = $33

 

       S + R = $43      [eq4]

       S       = $26      [eq5]

 --------------------    [Elimination; subtract equations]

             R = $17

 

Checking:

   Does    $26 + $30 + $33 + $17 = $108   ?

                            $108 = $108  ? yes

Answer 2

Hello Arvin

 

We have

 

S = amount Shelly receives

V = amount Veronica receives

C = amount Christy receives

R = amount Ruth receives

 

From the first clause we have

S + V + C + R = 106

 

Then

S + V = 56

S + C = 59

S + R = 43

 

Since S is in each of the 3 equations above, let's restate each equation for the other variable in terms of S.

 

S + V = 56      -->       V = 56 - S
S + C = 59      -->       C = 59 - S

S + R = 43      -->       R = 43 - S

 

Now we can substitute these expressions for V, C, and R into the first equation.

 

S + V + C + R = 106

 

Substituting

S + (56-S) + (59-S) + (43-S) = 106

S + 56 - S + 59 - S + 43 - S = 106

 

Combining terms

158 - 2S = 106

 

Rearrange terms

52 = 2S

26 = S

 

From here we can determine the amount each of the other girls receives.

 

V = 56 - S

V = 56-26=30

C = 59 - S

C = 59-26=33

 

R = 43 - S

R = 43-26=17

 

Shelly           $26

Veronica       $30

Christy         $33

Ruth            $17

              ---------

Total          $106