4 set Venn Diagram problem

Question

Trying to solve this for a few days. Couldnt solve it.

1.       Kroner asked 600 adults whether they had studied French, Arabic, German or Chinese in school. The report showed that:

215 had studied Chinese.

305 had studied German.

305 had studied French.

255 had studied Arabic.

90 had studied French and Chinese.

130 had studied French and Arabic.

150 had studied French and German.

110 had studied Arabic and German.

55 had studied Arabic and Chinese.

120 had studied German and Chinese.

40 had studied Chinese, Arabic and German.

60 had studied French, German and Chinese.

35 had studied French, Arabic and Chinese.

65 had studied French, Arabic and German.

Determine the numbers of people had studied all four kinds of languages

Answer 1

  

              [0.]      600 = A+B+C+D+E+F+G+H+I+J+K+L+M

1.	215 had studied Chinese.	215 = E+G+H+I+J+L+M
2	305 had studied German.	305 = D+F+G+I+J+K+L
3	305 had studied French.	305 = A+B+D+E+F+G+I
4	255 had studied Arabic.	255 = B+C+D+E+G+H+J
5	90 had studied French and Chinese.	90 = E+G+I
6	130 had studied French and Arabic.	130 = B+D+E+G
7	150 had studied French and German.	150 = D+F+G+I
8	110 had studied Arabic and German.	110 = D+G+J
9	55 had studied Arabic and Chinese.	55 = E+G+H+J
10	120 had studied German and Chinese.	120 = G+I+J+L
11	40 had studied Chinese, Arabic and German.	40 = G+J
12	60 had studied French, German and Chinese.	60 = G+I
13	35 had studied French, Arabic and Chinese.	35 = E+G
14	65 had studied French, Arabic and German.	65 = D+G
15	Determine the numbers of people had studied all four kinds of languages	?? = G

Now, how do you determine the value of G?

    If you know the values of I, J, F, or D, just use Equations 11-14.

    AND -- Once you know the value of G, you can compute values for I, J, F, and D to check.

    If you know the sum of (A+B+C+D+E+F+H+I+J+K+L+M), you can use Eq. 0 to find G.

Solutions sometimes seem to “go around Robin Hood’s barn,” but they get the result. So, we use that process.

   If we subtract Eq.12 and Eq.13 from Eq.5, we get:

              90 = E+G+I

            -60 =    -G-I

            -35 = -E-G

           ------------------

              -5 =   -G

               G = 5

You may continue to check the answer that G=5:

 

*** ERROR ***  Note that [5]-[12]-[13] gives G=-5 and with [14] D=60.

              G=6 with [16] and [17] gives D=70.

Answer 2

Chinese = c

German =g

French = f

Arabic = a

Find the number in each of these DISJOINT sets

Let v stand for intersection.

(cvgvfva)---- call this number x

(c'vgvfva) = 65-x

(cvg'vfva) = 65-x

(cvgvf'va)

(cvgvfva')

(c'vg'vfva)

(c'vgvf'va)

(c'vgvfva')

(cvg'vf'va)

(cvg'vfva')

(cvgvf'va')

(c'vg'vf'va)

(c'vg'vfva')

(c'vgvf'va')

(cvg'vf'va')

(c'vg'vf'va')