Topic 2 - 2.3  --  Problem Solving

One number is 9 more than 3 times another.  If their sum is 53.  What is the smaller number?

             x  =  smaller no.     This is called the legend.  Always set one of the

   3x  +  9  =  larger no.        unknowns equal to x or some other variable. Then

   x  +  3x  +  9  =  53           set the other unknown in terms of the same variable.

                                            The third step is the equation:  The sum of the two nos.

           4x   +  9  =  53           Now solve the equation.

                  4x    =  44

                    x    =  11           Now put x into the two unknowns at the top.

     x  =  smaller no.   11         You can check the problem by adding the two

  3x  +  9  =  larger no.            numbers (11  +  42  =  53).  Also 42 is 3 times

 3(11)  +  9  =  42                   11 plus 9.

The problem only asks for the smaller number (11), but I solved for both.

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The sum of three consecutive integers is 96.  What are the numbers?

             x  =  1st     Consecutive numbers are: 2,  3,  4,  x,  x  +  1, x  +  2

     x  +  1  =  2nd    Consecutive odd or even numbers are:  1, 3, 5 or 2, 4, 6

     x  +  2  =  3rd         x,  x  +  2, x  +  4

 x  +  x  +  1  + x  +  2  =  96           Sum -  add them together.

       3x  +  3  =  96                               x  =  1st      31

               3x  =  93                       x  +  1  =  2nd     32

                x  =   31                       x  +  2  =  3rd     33    

These are  three consecutive numbers and their sum is 96.

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The sum of the angle measures of a triangle is 180º.  The smallest angle in a triangle is 64 degrees less than the measure of the largest angle.  The measure of the remaining angle is 8 degrees more than the measure of the smallest angle.  What is the measure of each angle?

                    x  =  smallest angle       set x equal the smallest and the rest will be

          x  +  64  =  largest angle          based upon the smallest according to the

          x  +    8  =  other angle            problem.

        x  +  x  +  64  +  x  +  8  =  180     the equation (sum of the angles = 180º.

                 3x   +   72   =   180

                            3x    =   108

                              x    =   36

              x  =  smallest angle   36      The sum of these three angles is 180º.

    x  +  64  =  largest angle    100

    x  +    8  =  other angle        44  

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The length of a rectangle is 2 inches more than twice its width.  The perimeter of the rectangle is 28 inches.  What are the length and width of the rectangle?

It is a good idea to draw a rectangle here and label the:

      width  =  x

      length  =  2x  +  2

The formula for the perimeter of a rectangle is:  P  =  2L  +  2w.

Substituting into the formula:  28  =  2(2x  +  2)  +  2x

       28  =  4x  +  4  +  2x              The width is x which is   4

       28  =  6x  +  4                        The length is 2x  +  2 which is  10

       24  =  6x

        4  =  x

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The length of a rectangular floor is six times its width.  The perimeter of the floor is 210 feet.  What is the area of the floor?

  Two formulas are needed:   P  =  2L  +  2W    and   A  =  LW

   Width   =  x

   Length  =  6x

   210  =  2(6x)  +  2(x)            Substituting into the perimeter formula.

   210  =  12x     +  2x

   210  =  14x                         So the width (x) is 15 and the length (6x) is 90

     15  =    x                           Substituting into the area formula.

  A  =  (15)(90)

  A  =  1350  sq. ft.