Derfs and Enajs: Algebra and Venn Diagrams

Date: 03/09/2003 at 10:20:21
From: Ashley 
Subject: Derfs and Enajs

All Derfs are Enajs. One-third of all Enajs are Derfs. Half of all 
Sivads are Enajs. One Sivad is a Derf. Eight Sivads are Enajs. The 
number of Enjas is 90. How many Enajs are neither Derfs nor Sivads?

Date: 03/09/2003 at 15:04:09
From: Doctor Greenie
Subject: Re: Derfs and Enajs

Hello, Ashley  -

There are two basic approaches to a problem like this. One is purely 
algebraic: we write a series of equations based on the given 
information and try to solve that set of equations. The other is a 
visual process using Venn diagrams. I usually find the visual approach 
easier to use, but sometimes for a complex problem I resort to an 
algebraic approach. And sometimes using a combination of the two helps 
to keep track of what I am doing towards solving the problem.

So let's set up both the algebraic and visual approaches to this 
problem and see how we can use both of them to solve your particular 
problem.

For the visual approach, we use a Venn diagram, consisting of three 
mutually intersecting circles that represent the Enajs, Derfs, and 
Sivads.  Circles are hard to "draw" in typed text, so here is what my 
Venn diagram is going to look like:


                  EEEEEEEEEEEEEEEEEEEEEEEEE
                  E                       E
                  E   a                   E
                  E                       E
                  E       SSSSSSSSSSSSSSSS*SSSSSSSSS
                  E       S               E        S
                  E       S           e   E        S
                  E       S               E        S
          DDDDDDDD*DDDDDDD*DDDDDDDD       E        S
          D       E       S       D       E        S
          D       E   d   S   g   D       E        S
          D       E       S       D       E        S
          D       EEEEEEEE*EEEEEEE*EEEEEEEE        S
          D               S       D                S
          D               S   f   D           c    S
          D               S       D                S
          D       b       SSSSSSSS*SSSSSSSSSSSSSSSSS
          D                       D
          D                       D
          D                       D
          DDDDDDDDDDDDDDDDDDDDDDDDD

The "E" box represents all the Enajs; the "D" box represents all the 
Derfs; and the "S" box represents all the Sivads.  These boxes all 
intersect, giving us seven regions, which I have labeled with lower-
case letters as follows:

  a:  Enaj yes; Derf  no; Sivad  no
  b:  Enaj  no; Derf yes; Sivad  no
  c:  Enaj  no; Derf  no; Sivad yes
  d:  Enaj yes; Derf yes; Sivad  no
  e:  Enaj yes; Derf  no; Sivad yes
  f:  Enaj  no; Derf yes; Sivad yes
  g:  Enaj yes; Derf yes; Sivad yes

For example, the region labeled "e" is inside the "E" and "S" boxes 
but outside the "D" box; so any element in this region is both an 
Enaj and a Sivad but is not a Derf.

To set up the algebraic approach, I define variables to represent the 
numbers of elements that possess the different possible combinations 
of characteristics. I come up with a list identical to the preceding 
list:

  let a = number which are Enajs but not Derfs or Sivads
  let b = number which are Derfs but not Enajs or Sivads
  ...
  let g = number which are Enajs and Derfs and Sivads

Now we are ready to solve the problem by using the given information 
both in the Venn diagram and in our algebraic approach.

(1) All Derfs are Enajs

In the Venn diagram, this means there are no elements in the "D" box 
that are outside the "E" box; so there are no elements in regions b 
or f.

Algebraically, we have

  b=0
  f=0

(2) One-third of all Enajs are Derfs

We can't do a lot with this information just yet in our Venn diagram 
(I use it in step (1) below). This information tells us that the 
combined number of elements in regions d and g (the elements that are 
both Enajs and Derfs) is one-third the combined number of regions 
a, d, e, and g (the total number of Enajs).

Algebraically, we have

  d+g = (a+d+e+g)/3

This piece of information is equivalent to saying that there are twice 
as many elements in regions a and e together as there are in regions d 
and g together; so the above algebraic equation is equivalent to the 
following:

  a+e = 2(d+g)

(3) Half of all Sivads are Enajs

Again, we can't do a lot with this information just yet in our Venn 
diagram; this information tells us that the combined number of 
elements in regions e and g (the elements that are both Enajs and 
Sivads) is one-half the combined number of regions c, e, f, and g 
(the total number of Sivads).

Algebraically, we have

  e+g = (c+e+f+g)/2

This piece of information is equivalent to saying that there are as 
many elements in regions e and g together as there are in regions c 
and f together; so the above algebraic equation is equivalent to the 
following:

  c+f = e+g

(4) One Sivad is a Derf

In the Venn diagram, this tells us that the combined number of 
elements in regions f and g is 1.

Algebraically, we have

  f+g = 1

(5) Eight Sivads are Enajs

In the Venn diagram, this tells us that the combined number of 
elements in regions e and g is 8.

Algebraically, we have

  e+g = 8

(6) The number of Enajs is 90

In the Venn diagram, this information tells us that the combined 
number of elements in regions a, d, e, and g (representing all the 
Enajs) is 90.

Algebraically, we have

  a+d+e+g = 90

Now we are ready to combine the pieces of information we have to find 
the numbers of elements in each region in the Venn diagram - i.e., to 
find the values of our algebraic variables a, b, c, d, e, f, and g.

We currently have the following:

(1) b = 0  and  f = 0
(2) a+e = 2(d+g)
(3) c+f = e+g
(4) f+g = 1
(5) e+g = 8
(6) a+d+e+g = 90

Combining (1) and (4), we find

(7) g = 1

(I am giving the explanation algebraically because it is easier in a 
typed explanation; in practice, I look at the Venn diagram and see 
there are no elements in region f and a total of 1 element in regions 
f and g together, so I conclude that there is 1 element in region g.)

Then combining (5) and (7), we find

(8) e = 7

(Again the given explanation is algebraic, but in practice I reach 
this conclusion by looking at my Venn diagram.)

Next, combining (3) with (1), (7), and (8), we find

(9) c = 8

The nature of the given information makes steps (7) through (9) above 
fairly obvious - they are conclusions which are easily drawn from the 
given information either algebraically or using the Venn diagrams.

From this point on, however, there are various different paths to the 
final solution. The one that I think is easiest is apparent (to me) 
with the Venn diagram but rather difficult to see with the algebraic 
approach. So let me make my explanation using the Venn diagram instead 
of the algebraic approach.

Item (6) tells me that the total number of Enajs is 90; item (2) tells 
me that there are twice as many Enajs that are NOT Derfs as there are 
Enajs that are Derfs. From this I conclude

(10)  a+e = 60  and  d+g = 30

Then I finish the problem by combining (10) with (8) to get

(11) a = 53

and by combining (10) with (7) to get

(12) d = 29

When I'm all done, as a check of the work I have done, I label each 
region of my Venn diagram with the numbers of elements I have 
determined for each region and go back and verify that those numbers 
fit all the given information. Again, I find this check easier to do 
using the Venn diagram than algebraically.

And of course, having done all that work, we must remember to answer 
the question, which is, "How many Enajs are neither Derfs nor 
Sivads?" The answer is the number of elements in region a, which is 
53.

I hope all this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/12/2003 at 17:34:36
From: Ashley 
Subject: Thank you (Derfs and Enajs)

Dear Dr. Math,

Thanks a lot for helping me solve my word problems. They are my 
toughest part of math and I really appreciate your help. I hope you 
continue to help me out more this school year.

Ashley