Math Help - Venn Diagram

1. Venn Diagram

I have been trying to solve this problem for two day but i really can't find the answer. can you help me?

Problem:

Rye has 16 suitors
8 are tall
9 are dark
5 don't have the qualities of being tall nor handsome
8 have atleast 2 qualities
10 are ugly
1 is tall but not dark nor handsome.
3 are dark but not tall nor handsome.

2. Re: Venn Diagram

What is the question?

3. Re: Venn Diagram

The questions are:
How many are neither tall nor dark, nor handsome?
If Rye goes for physical appearance, will it be hard for her to choose among her suitors?

And how can i get the Venn Diagram summary of it?
Thanks!

4. Re: Venn Diagram

You will have three circles. Let's label the quantities. There are 8 disjoint regions in a 3-variable Venn diagram.

Let A be those suitors who are tall but not dark or handsome.
Let B be those suitors who are both tall and dark but not handsome.
Let C be those suitors who are dark but neither tall nor handsome.
Let D be those suitors who are tall and handsome but not dark.
Let E be those suitors who are tall, dark, and handsome.
Let F be those suitors who are dark and handsome but not tall.
Let G be those suitors who are handsome but neither tall nor dark.
Finally, let H be those suitors who are neither tall, dark, nor handsome.

Your 8 statements all translate into an equation you can write down.

There are 8 tall suitors: A+B+D+E = 8.
There are 16 suitors: C+F+G+H = 8 (in other words, all the suitors not mentioned above).
There are 9 dark suitors: B+C+E+F = 9.

Can you write the rest of the equations down? You need 8 linearly independent equations.

5. Re: Venn Diagram

Thank you so much.. this will help me a lot^^

6. Re: Venn Diagram

You're welcome! Let me know if you have any further hang-ups.

7. Re: Venn Diagram

I have solved the Venn diagram and I used the 8 equations to find the answer through trial and error. But we had a long exam and it took me a lot of time to solve another venn diagram problem. I ran out of time and so i was not able to complete it. is there any way i can use the equations without doing trials and errors?

8. Re: Venn Diagram

You should become very familiar with Gaussian Elimination with Back Substitution. It is the fastest known exact method, in general, to solve a linear system of equations. It's certainly faster than trial-and-error. However, with the system you have for this problem, sometimes a heuristic look at some judicious substitutions can be a bit faster. That's what I did, primarily because the coefficient matrix for this system is fairly sparse (meaning, there are a lot of zero entries). If I recall correctly, there were two variables that I could just write down, and that sort of "cascaded" the implications for what the other variables had to be.

9. Re: Venn Diagram

Ok then i just have to be familiar with Gaussian Elimination with back substitution.. thanks again

10. Re: Venn Diagram

You're welcome!

11. Re: Venn Diagram

Hello, nickgc!

The problem is much easier than you think.
. . Most of the information is unnecessary.

(1) Rye has 16 suitors
(2) 8 are tall
(3) 9 are dark
(4) 5 don't have the qualities of being tall nor handsome
(5) 8 have at least 2 qualities
(6) 10 are ugly
(7) 1 is tall but not dark nor handsome.
(8) 3 are dark but not tall nor handsome.

How many are neither tall, dark nor handsome?

We have this three-ring Venn diagram.
. . And we want to determine $b.$

Code:
      *---------------------------*
|                           |
|   *---------------*       |
|   | Tall          |       |
|   |       *-------+---*   |
|   |       |       |   |   |
|   |   *---+---*   |   |   |
|   |   |   |   |   | a |   |
|   *---+---+---+---*   |   |
|       |   |   |       |   |
|       |   |   |  Dark |   |
|       |   *---+-------*   |
|       | Hsome |           |
|       *-------*       b   |
|                           |
*---------------------------*

From statement (8): $a = 3$

From statement (4): $a + b \,=\,5$

. . Therefore: $b = 2$

12. Re: Venn Diagram

Originally Posted by Soroban
Hello, nickgc!

The problem is much easier than you think.
. . Most of the information is unnecessary.

We have this three-ring Venn diagram.
. . And we want to determine $b.$

Code:
      *---------------------------*
|                           |
|   *---------------*       |
|   | Tall          |       |
|   |       *-------+---*   |
|   |       |       |   |   |
|   |   *---+---*   |   |   |
|   |   |   |   |   | a |   |
|   *---+---+---+---*   |   |
|       |   |   |       |   |
|       |   |   |  Dark |   |
|       |   *---+-------*   |
|       | Hsome |           |
|       *-------*       b   |
|                           |
*---------------------------*

From statement (8): $a = 3$

From statement (4): $a + b \,=\,5$

. . Therefore: $b = 2$

Yes, that will certainly answer the first question adequately and quickly. What about the second question (in post # 3)?

13. Re: Venn Diagram

You could also consider filling in as many values as possible
in the Venn diagram first,
then writing equations for the remaining values
that will enable you to answer the question.

The final 2 clues allows 2 regions of the Venn diagram to be filled in.
Next, one of the remaining clues allows us to find one of x or y
in the attachment.

Then, another clue allows us to find the other of these 2 values.

Finally, it's possible to write 3 equations to solve for e, f and g.

14. Re: Venn Diagram

Okay, here's the full treatment.
(Archie Meade beat me to it . . .)

(1) Rye has 16 suitors
(2) 8 are tall
(3) 9 are dark
(4) 5 are not tall nor handsome
(5) 8 have at least 2 qualities
(6) 10 are not handsome
(7) 1 is tall but not dark nor handsome.
(8) 3 are dark but not tall nor handsome.

How many are neither tall, dark nor handsome?

If Rye goes for physical appearance, will it be hard to choose among her suitors?

We have this three-ring Venn diagram.

Code:
      *---------------------------*
|                           |
|   *---------------*       |
|   | Tall          |       |
|   |  a    *-------+---*   |
|   |       |   b   |   |   |
|   |   *---+---*   | c |   |
|   |   | d | e |   |   |   |
|   *---+---+---+---*   |   |
|       |   |   |       |   |
|       | f | g |  Dark |   |
|       |   *---+-------*   |
|       | Hsome |           |
|       *-------*       h   |
|                           |
*---------------------------*

From statement (7): $a = 1$
From statement (8): $c = 3$
From statement (4): $c+h \,=\,5 \;\;\Rightarrow\;\; h = 2$
From statement (6): $a + b + c + h \,=\,10 \;\;\Rightarrow\;\; 1+b+3+2\,=\10 \;\;\Rightarrow\;\; b = 4$

The diagram becomes:

Code:
      *---------------------------*
|                           |
|   *---------------*       |
|   | Tall          |       |
|   |  1    *-------+---*   |
|   |       |   4   |   |   |
|   |   *---+---*   | 3 |   |
|   |   | d | e |   |   |   |
|   *---+---+---+---*   |   |
|       |   |   |       |   |
|       | f | g |  Dark |   |
|       |   *---+-------*   |
|       | Hsome |           |
|       *-------*       2   |
|                           |
*---------------------------*

$\begin{array}{ccccccccc}\text{From statement (2):} & d + e + 1 + 4 \:=\:8 & \Rightarrow & d + e &=& 3 & (a) \\ \text{From statement (3):} & e + f + 4 + 3 \:=\:9 & \Rightarrow & e + f &=& 2 & (b) \\ \text{From statement (5):} & d + e + f + 4 \:=\:8 & \Rightarrow & d + e + f &=& 4 & (c) \end{array}$

$\begin{array}{ccccccc}\text{Substitute (a) into (c):} & 3 + f \:=\:4 & \Rightarrow & f \:=\:1 \\ \text{Substitute (b) into (c):} & d + 2 \:=\:4 & \Rightarrow & d \:=\:2 \end{array}$

And hence: . $e \:=\:1\,\text{ and }\,g \:=\:2$

And the final diagram is:

Code:
      *---------------------------*
|                           |
|   *---------------*       |
|   | Tall          |       |
|   |  1    *-------+---*   |
|   |       |   4   |   |   |
|   |   *---+---*   | 3 |   |
|   |   | 2 | 1 |   |   |   |
|   *---+---+---+---*   |   |
|       |   |   |       |   |
|       | 2 | 1 |  Dark |   |
|       |   *---+-------*   |
|       | Hsome |           |
|       *-------*       2   |
|                           |
*---------------------------*

Define "hard to choose".

Among the 16 suitors, only 2 are short, light and ugly.
She has 14 chances out of 16 to seclecting a suitor
. . who is has at least one attractive characteristic.

If she insists on Perfection, only one suitor is tall and dark and handsome.

15. Re: Venn Diagram

so all i just have to do are judicious substitutions for instances like this..
thanks to all of you ^^