A survey of the last 105 cars that have been brought to an automobile dealership for repair produced the following data.

Question

A survey of the last 105 cars that have been brought to an automobile dealership for repair produced the following data.
28 had brake repairs 8 had brake and exhaust repairs
21 had exhaust repairs 6 had brake and transmission repairs
13 had transmission repairs 3 had exhaust and transmission repairs

If 1 cars had all three repairs performed, How many had...
exactly one of these three repairs done
none of these three repairs done
exactly two of these three repairs done

Answer 1

Hi Sky,

 

When expressing various set combinations, it is best to express visually what is presented verbally.  Set combinations are very nicely drawn out via a Venn diagram (overlapping circles) with numbers filled in to represent the various set combination quantities.

 

Click this link (https://dl.dropbox.com/s/1eb8oxzgiviv2g3/Venn_Diagram.png?raw=1) to view our Venn diagram pertaining to this problem for reference in our solution below.

 

The three circles represent the three kinds of repairs (<B>rake, <E>xhaust, <T>ransmission).

 

The red number represents the 1 repair job requiring all three repairs.  The blue numbers represent the two repair job combinations.  The red and blue numbers were filled in from the given information in the problem.  

 

The green numbers represent the single repair jobs and were calculated as follows:

 

    Single Brake Jobs = Total Brake Jobs - 

                                  Combination  Break Jobs

                              = 28 - (8 + 1 + 6)

                              = 28 - 15

                              = 13

 

    Single Exhaust Jobs = Total Exhaust Jobs -

                                     Combination Break Jobs

                                 = 21 - (8 + 1 + 3)

                                 = 21 - 12

                                 = 9

 

    Single Trans. Jobs = Total Trans. Jobs -

                                  Combination Trans. Jobs

                               = 13 - (6 + 1 + 3)

                               = 13 - 10

                               = 3

 

Now for the answers to our three questions:

 

    Exactly 1 of 3 Repairs = Sum of Green Numbers

                                        (Single Jobs)

                                    = 15 + 9 + 3

                                    = 27

   

    Exactly 2 of 3 Repairs = Sum of Blue Numbers

                                       (Double Combinations)

                                    = 8 + 6 + 3

                                    = 17

 

    None of 3 Repairs =

          Total Repair Jobs  

        - Sum of Green Numbers (1 repair jobs)

        - Sum of Blue Numbers (2 repair jobs)

        - Red Number (3 repair jobs)

    None of 3 Repairs = 105 - 27 - 17 - 1

                             = 60

 

The use of a Venn diagram in the representation of set information is an ideal tool for answering set combination questions.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.