Solving Venn Diagrams.ppt - MathXTC

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							           SOLVING
        VENN DIAGRAMS




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The universal Set, U, and the sets A,
B and Contained within it are such
that:
 n(U )  74     n( A  B)  10
 n( A)  25     n( B  C )  11
 n( B )  34    n( A  C )  16
 n(C )  37     n( A  B  C )  8


Find: n( A  B  C )
           SOLVING
        VENN DIAGRAMS

          Method 1:
      Algebraic Approach


www.mathxtc.com
The universal Set, U, and the sets A,
B and Contained within it are such
that:
 n(U )  74
                     A                 B
                  n( A  B)  10
 n( A)  25       n( B  C )  11
 n( B )  34      n( A  C )  16
 n(C )  37       n( A  B  C )  8


Find:
 n( A  B  C )                C
The universal Set, U, and the sets A,
B and Contained within it are such
that:                                      74


 n(U )  74
                     A                 B
                  n( A  B)  10
 n( A)  25       n( B  C )  11
 n( B )  34      n( A  C )  16
 n(C )  37       n( A  B  C )  8


Find:
 n( A  B  C )                C
The universal Set, U, and the sets A,
B and Contained within it are such
that:                                            74


 n(U )  74
                     A                       B
                  n( A  B)  10
                       25               34
 n( A)  25       n( B  C )  11
 n( B )  34      n( A  C )  16
 n(C )  37       n( A  B  C )  8


Find:                              37

 n( A  B  C )                C
 The universal Set, U, and the sets A,
 B and Contained within it are such
 that:                                                    74


n( A  B)  10
                     A                                B

                                       10
                         25                      34
n( B  C )  11
n( A  C )  16
n( A  B  C )  8            16            11




  Find:                                37

   n( A  B  C )                  C
 The universal Set, U, and the sets A,
 B and Contained within it are such
 that:                                                    74


n( A  B)  10
                     A                                B

                                       10
                         25                      34
n( B  C )  11
n( A  C )  16
n( A  B  C )  8            16            11




  Find:                                37

   n( A  B  C )                  C                      8
 The universal Set, U, and the sets A,
 B and Contained within it are such
 that:                                                    74


n( A  B)  10
                     A                                B

                                       10
                         25                      34
n( B  C )  11
n( A  C )  16
                                       x
n( A  B  C )  8            16            11




  Find:                                37

   n( A  B  C )                  C                      8
                                                            74

A                                                       B

                            10
    25                                             34
                       10 - x
          a                               b


                            x
              16 - x             11 - x
         16                                   11

                            c



                            37
                        C                                   8
a + 10 + (16 – x) = 25                                                              74

a=x–1                   A                                                      B

                                                    10
                             25                                            34
                                                    10-x
                                   -
                                  xa 1                            b


                                                     x
                                         16-x              11-x
                                  16                                  11

                                                    c



                                                    37
                                                C                                   8
a + 10 + (16 – x) = 25                                                              74

a=x–1                   A                                                      B
b + 10 + (11 – x) = 34                              10
                             25                                            34
 b = x + 13
                                                    10-x
                                                                     b
                                                                  x + 13
                                   x-1

                                                     x
                                         16-x              11-x
                                  16                                 11

                                                    c



                                                    37
                                                C                                   8
 a + 10 + (16 – x) = 25                                                                  74

 a=x–1                         A                                                    B
b + 10 + (11 – x) = 34                                     10
                                    25                                          34
 b = x + 13
                                                           10-x
                                          x-1                            x+13

 n( A  B  C )  7                                         x
                                                16-x              11-x
                                         16                                11
                                                            c



                                                           37
37 + (x – 1) + (10 – x) + (x + 13) = 74 – 8
                                                       C                                 8
59 + x = 66
x=7
 a + 10 + (16 – x) = 25                                                           74

 a=x–1                         A                                             B
b + 10 + (11 – x) = 34                                    10
                                    25                                   34
 b = x + 13
                                                          3
                                              6                    20

 n( A  B  C )  7                                       7
                                                  9            4
                                         16                         11
                                                          17



                                                          37
37 + (x – 1) + (10 – x) + (x + 13) = 74 – 8
                                                      C                           8
59 + x = 66
x=7
           SOLVING
        VENN DIAGRAMS

           Method 2:
     Trial and Adjustment
           Approach

www.mathxtc.com
Here is the given information:
                                                      74

                 A                                B

                                   10
                     25                      34




                          16            11

Find:
n( A  B  C )
                                   37
                               C                      8
              n( A  B  C )  10            74

A                                        B

                        10
    25                              34




         16                   11




                        37
                    C                        8
Try x = 4 ?                                                               7474
                 AA                                                  BB
n(U )  71
                                              10
                      2525                                    3434
Needs to be 74
                                              6
                                3                      17

                                              4
                                    12             7
                             1616                      1111
                                              14



                                              37
                                              37
                                         CC                               88
Now Try x = 7 ?                                                           7474
                  AA                                                 BB
n(U )  74
                                              10
                       2525                                   3434
Bingo !!!
                                              3
 x=7                             6                     20
74 – 71 = 3 and                               7
4+3=7                                9             4
Is this a                     1616                     1111
coincidence? 
                                              17



                                              37
                                              37
                                         CC                               88
           SOLVING
        VENN DIAGRAMS

         Method 3:
     Reasoning Approach


www.mathxtc.com
Here is the given information:
                                                      74

                 A                                B

                                   10
                     25                      34




                          16            11

Find:
n( A  B  C )
                                   37
                               C                      8
                                                                 74
25 + 34 + 37 = 96
                            A                                B
A + B + C = 66
                                              10
                                25                      34
96 – 66 = 30 Surplus

10 + 16 + 11 = 37

37 – 30 = 7 Surplus                           7

                                     16            11

Find:
n( A  B  C )         =7

                                              37
                                          C                      8