Very Elementary Venn Diagram

[speterson]

My daughter has an assigment that I just can't figure out.

Problem: A group of 20 children went to the amusement park. 6 children rode only the roller coaster and log flume. 2 rode only the roller coaster and sky buckets. 3 rode only the log flume and sky buckets. 4 rode only the sky buckets. Only one child rode all three rides. Ten children rode the roller coaster. How many children rode only the log flume?

I have not done Venn diagrams in forever so I am unable to help her. I got 15 as the answer, but who knows. Any help would be great!

 

[arthur ohlsten]

I believe the answer is 3

total children=20
let R be the number of children that rode the roller coaster
let S be sky buckets " " " " " sky bucket
let L be log fume " " " " " log fume

given:
RL=6
RS=2
LS=3
S=4
RLS=1
R =10

sketch 3 0ne inch circles that overlab giving 7 unique areas. Mark on the outside of the 3 circles R or S or L defining one circle for each ride.

where all 3 overlap [RLS} place a 1.

in the area where R and L overlap but not where all three overlap place a 6.
or we have 6 children rode only the R,and L

in the area where R and S have in common, but not where all 3 intercept place a
2 or we haverode the R and S

in the area where the L and S intercect, but not LRS< place a 3

in the S circle , outside the intersections of the other two circles but within S , place a 4. Or 4 children only rode the S

now ten children rode the R, but we have a total of 9 children that rode the R and the L or S or both. So in the R circle place a 1.

we now have accounted for 17 children, so L alone must have 3
s place in the L circle but not in any of its intersections a 3 ANSWER

Arthur
please check my work.

 

[soroban]

Hello, speterson!

Arthur did an excellent job!

Here are the accompanying Venn diagrams.
[Don't mind me . . . I love the challenge of typing pictures.]

[1] A group of 20 children went to the amusement park.
[2] 6 children rode only the Roller coaster and Log flume.
[3] 2 rode only the Roller coaster and Sky buckets.
[4] 3 rode only the Log flume and Sky buckets.
[5] 4 rode only the Sky buckets.
[6] Only one child rode all three rides.
[7] Ten children rode the Roller coaster.
How many children rode only the Log flume?

Here is the basic Venn diagram.
There are three "circles" and seven regions in the diagram.
. . \(\displaystyle R\) = Roller coaster, \(\displaystyle \;L\) = Log flume, \(\displaystyle \;S\) = Sky bucket.

        R *-----------* 
          | f         |
          |   *-------+---* L
          |   |    d  |   |
      *---+---+---*   |   |
      |   | c | a |   |   |
      |   *---+---+---*   |
      |       | b |    g  |
      |       *---+-------*
      |   e       |
    S *-----------*

From [6], we have: \(\displaystyle \,a\,=\,1\)

        R *-----------* 
          | f         |
          |   *-------+---* L
          |   |    d  |   |
      *---+---+---*   |   |
      |   | c | 1 |   |   |
      |   *---+---+---*   |
      |       | b |    g  |
      |       *---+-------*
      |   e       |
    S *-----------*

From [4]: \(\displaystyle b \,=\,3\)
From [3]: \(\displaystyle c\,=\,2\)
From [2]: \(\displaystyle d\,=\,6\)
From [5]: \(\displaystyle e \,=\,4\)

        R *-----------* 
          | f         |
          |   *-------+---* L
          |   |    6  |   |
      *---+---+---*   |   |
      |   | 2 | 1 |   |   |
      |   *---+---+---*   |
      |       | 3 |    g  |
      |       *---+-------*
      |   4       |
    S *-----------*

[3] says 10 children were in \(\displaystyle R.\)
We have accounted for: \(\displaystyle 2\,+\,1\,+\,6\:=\:9\) of them.
Hence: \(\displaystyle f\,=\,1\)

        R *-----------* 
          | 1         |
          |   *-------+---* L
          |   |    6  |   |
      *---+---+---*   |   |
      |   | 2 | 1 |   |   |
      |   *---+---+---*   |
      |       | 3 |    g  |
      |       *---+-------*
      |   4       |
    S *-----------*

We have accounted for: \(\displaystyle 1\,+\,6\,+\,2\,+\,1\,+\,4\,+\,3 \:=\:17\) children.

[1] says there are 20 children. .Hence: \(\displaystyle g\,=\,3\)


Therefore, 3 children rode only the Log Flume.

 

[arthur ohlsten]

I must learn how to draw circles or intersecting squares. It makes the answers much easier to "see".
Arthur

 

[stapel]

My daughter has an assigment that I just can't figure out....

Surely you aren't doing her homework for her...? :shock:

Eliz.

 

[speterson]

Thanks to all! It was a great help.

And no I wasnt doing her homework for her, but how can I show her how to do it unless I know how?!?! How does that question help me?

 

[stapel]

When a student is working on the assignment and has a question, the student generally asks the question. That is to say, usually the one doing the work is the one doing the asking. So the question was fairly straightforward.

Meanwhile, since (you indicate) you do not understand the material yourself, attempting to tutor "through" you is, I'm sorry to say, unlikely to be successful. If the student has difficulty on the next exercise of this type, please have the student post (or dictate a post), showing what she had tried and clearly stating where she is stuck. In this way, the tutors will be able to "see" where she is having trouble, and will be able to better provide long-term help.

Thank you for your understanding.

Eliz.

P.S. Note: Anyone who has tutored in this sort of environment has encountered many posts of this type. (You couldn't be expected to know this, which is why I take the time to explain the situation.) The poster claims to be the parent, and wants the complete worked solution in order to somehow teach the material to his child. Of course, if the entire chapter could have been explained from one worked solution, the chapter would have been much shorter! *smile* All too often, the "parent" turns out, in reality, to be the student. As you are a parent yourself, I'm sure you can understand why most responsible tutors would want to exercise sensible caution. I apologize for any confusion.

 

[speterson]

Who are you? She is 10 years old and all I asked for was help not a lesson from someone who obviously has nothing better to do than read everyone else's post. If you don't feel like helping then move on...I got plenty of answers from other people, you apparently have nothing else to offer besides pointless advice. I have been a mother for a long time and am doing just fine thanks! GAL...