Hints for solving the problems - Apply 2.3

In this exercise you will be solving word problems. The following steps may help you set them up.

1.  Read the problem.

2.  Represent the unknown as a variable. All unknowns should be listed

         using the same variable. (called the legend)

3.  Set up an equation that describes the problem.

4.  Solve the equation.

5.  Check the answers to see if they satisfy the problem.

Problem No. 2 – The sum of two numbers is 43. One number plus three times the other number is 65. What are the numbers?

   Let x = one of the numbers.

   Then 43 - x is the other number.    Step No. 2 from solving word problems.

Remember is you know the sum of two numbers, you can take one number and subtract it from the sum and get the other number.

The problem says one number plus three times the other number is 65. It is easier to take 3 times x than 3 times (43 - x).

        43  - x  +  3x  =  65

       -43                      -43      Subtracting 45 from both sides of equation.

                     2x     =    22

                      2             2        Dividing by 2

                        x    =    11

       So one of the numbers is 11.

       The other number is 43 – 11 = 32     11 & 32 are the answers

   It is a good idea to check your answers.

    32 + 3(11) = 65 32 + 33 = 65 We have correct solutions.

Problem No. 12 – The sum of three consecutive even integers is 444. What are the numbers?

To start consecutive number problem, you need to know the following:

    3 consecutive numbers – 1, 2, 3

    3 consecutive even numbers – 2, 4, 6 

    3 consecutive odd numbers – 1, 3, 5

When you start any consecutive number problem, always start with x.

  3 consecutive nos. 3 consecutive even nos. 3 consecutive odd nos.

         x, x + 1, x + 2           x, x + 2, x + 4                      x, x + 2, x + 4

For this problem the legend is:     x           1^st number

                                                      x + 2         2^nd number

                                                      x + 4          3^rd number

                      x   +   x + 2   +   x + 4   =   444     the equation

                              3x   +   6   =   444                  solving

                                    3x    =    438

                                      x    =    146

                     So the 1^st answer is:    x  =   146

                         The 2^nd answer is:   x + 2  = 148

                         The 3^rd answer is:    x + 4  =   150

       Checking: Does 146 + 148 + 150 = 444? yes, so the answers are:

                                      146, 148, 150

Problem No. 18 – Carl is 9 years older than his cousin Jenny. If the sum of their ages is 77, how old is each one of them?

      Let   x = Jenny’s age

     Then   x + 9 = Carl’s age. The problem says Carl is 9 years older than Jenny.

      Then   x  +  x + 9  =  77       The sum of their ages is 77.

                       2x + 9   =  77

                              2x  =   68

                                x  =    34

          So Jenny’s age is:   x  =   34

           And Carl’s age is:     x + 9 =  43

    Does the sum of their ages equal to 77? 34 + 73 = 77 Yes

                  The answers are: Jenny 34 & Carl  43.

Problem No. 26 – Gerhard is twice as old as Isolde. Sixteen years ago, Gerhard was four times as old as Isolde was sixteen years ago. How old is each one now?

     Insole’s age now:   x      Insole’s age 16 years ago:   x - 16

  Gerhard’s age now:   2x   Gerhard’s age 16 years ago:   2x - 16

      2x - 16 = 4(x - 16) 16 years ago Gerhard was 4 times as old as Isolde, so to set up an equation you must multiply Insole’s age by 4 to make them equal.

                 2x - 16 = 4(x - 16)

                 2x - 16 = 4x - 64

                     2x    = 4x - 48

                      -2x  =   - 48

                          x  =   24

                So Isolde now is:   x = 24

        And Gerhard now is:   2x =  48

          Checking: 24 - 16 = 8

                             48 - 16 = 32

             16 years ago was Gerhard 4 times as old as Isolde? yes

                 So the answers are: Isolde 24 &Gerhard 48

Problem No. 30 – If the largest angle of an isosceles triangle measure 68 degrees, what are the measures of the other two equal angles?

An isosceles triangle has two equal angles and two equal sides.

Also the sum of the angles of any triangle is 180 degrees.

Remember the perimeter is the sum of all three sides.

   Let x = the measure of the two equal angles.

      Then x + x + 68 = 180

                    2x + 68 = 180

                         2x    = 112

                           x     =  56

    So each of the two equal angles is 56 degrees

Problem No. 36 – The longest side of a triangle is 7 cm longer than the shortest side. The remaining side is 3 cm shorter than the longest side. The perimeter of the triangle is 29 cm. What is the length of each side?

   Let the shortest side   =   x

    Then the longest side =   x  +  7

The remaining side =  x  +  4 The problem says the remaining side is 3 cm shorter than the longest side, so subtract 3 from x + 7.

            x  +  x + 7  +  x + 4  =  29       the equation

                      3x    +    11    =   29

                               3x         = 18

                                    x     =    6

      So the shortest side is:    x  =  6

   Then the longest side is:   x   +   7  =   13

        The remaining side is:   x   +   4  =   10

    Does the sum of the three sides equal to 29? 6 + 13 + 10 = 29 yes

        So the shortest side is: 6

                   Longest side is: 13

                Remaining side is: 10

Problem No. 46 – The width of a rectangle is 52 inches less than four times its length. The perimeter of the rectangle is 51 inches. What are the length and width of the rectangle?

When solving these Geometry problems it is helpful to draw the rectangle and put the value of the length and width on the sides.

    Let the length   =   x

   And the width    =   4x   -   52 The problem states that the width is 52 inches less than  four times its length.

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

                  2x  +  2(4x  -  52)  =  51          the equation

                    2x  +   8x   -  104 =  51

                          10x      -  104 =  51

                                10x   =   155

                                    x    =   15.5

                   So the length is:   x  =   15.5

                  And the width is:   4x   -   52      4(15.5)  -  52  =  10

    Checking using the perimeter formula 2(15.5) + 2(10) = 51

    The answers are: width 15.5     length 10