Venn Diagrams

Question

Hi there,

Parks a survey of 300 parks showed the following:

 

15 had only camping

20 had only hiking trails

35 had only picnicking

185 had camping

140 had camping and hiking trails

125 Had camping and picnicking

210 had hicking trails

 

Determine the number of parks that

1. Didn't have any of these features

2. Had exactly two of these features

Answer 1

Make 2 circles, partially overlapping, near the top of a page.  Then make a third circle below both of these, overlapping both.  The top two circles should be labeled C and H, from left to right (put these capital letters on the circumference, at top). Label the lower circle P (on circumference, at bottom).

 

Circle C can be viewed as having four distinct areas:  Along top, from left to right:  15 & b

                                                                             Beneath those, also L to R: a & c.

15 tells us that there are 15 parks having camping only; there are b parks having both Camping & Hiking

                                                                             Beneath: a have Camping & Picknicking; c have all three (P, C, H)

In the upper right corner of the H circle, put 20 (20 parks have only Hiking); beneath that, place letter d in that region representing parks having Hiking and Picknicking.

 

Lastly, place 35 in the bottom part of circle P, for 35 parks having Picknicking only.

 

Now, if your picture is correct (i.e. if my instructions make sense to you), the following 4 given statements can be turned into equations:

 

185 had camping                            185 = 15 + a +b +c
140 had camping and hiking trails    140 = b + c
125 Had camping and picnicking      125 = a + c
210 had hiking trails                      210 = 20 + b + c + d

 

This linear system of 4 equations in four unknowns can be solved for a, b,c & d.  Personally, I like to use the TI-83/84 MATRIX menu's RREF function on the equivalent Augmented Matrix.  But you can use any method known to you.

 

 

 

Answer 2

Here is a rather crude attempt at a Venn Diagram (imagine interconnecting circles around these):

 

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

     e others

 

            C                                               H

                15                   a                20

 

                                      b

 

                          c                   d

 

                                      35

                                       P

 

Put in 15, 20, and 35.

185 = 15 + a + b + c              [eq1]

140 = 15 + 20 + a + b           [eq2]

125 = 15 + 35 + b + c            [eq3]

210 = 20 + a + b + d              [eq4]

 

Rewrite eq2 as (a+b = 105) and plug into eq1 (185 = 15 + 105 + c) to get c=65.

 

Put c=65 into eq3 (125 = 15+35+b+65) to get b=10.

 

Put those into eq1 (185 = 15 + a  + 10 + 65) to get a=95.

 

And, finally eq4, (210 = 20 + 95 + 10 + d) to get d=85.

 

Add the values in each area and subtract from 300 total:

     e = 300 – (15+95+20+10+65+85+35) = 300 – 325

      Oops, either I have a calculation error or you made a typo.

     Anyway, this is the process.   You can do it!!

 

Q1:  use the value of e

Q2:  add  a+c+d