How to Solve this with a Venn diagram?

Question

The problem is that:

32 people have scheduled a certain math class.

28 people have scheduled a certain science class.

32 people have scheduled a certain composition class.

the math and science class have 19 people in common.

the composition and science class have 21 people in common.

the composition and math class have 15 people in common.

all three classes have 12 people in common.

75 people were asked what their schedules contained.

Answer 1

go to this site:

http://interactives.mped.org/view_intera...

it will give you a good image to start with

but here is a start:

12 people have all three classes

the math and science classes also have (19-12) people in common who don't take the composition class

the composition and science class have (21-12) people in common who don't take the math class

and the composition and math class have (15-12) people in common who don't take the science class

so you have 12 people who take all three classes

7 people who take the math and science only

11people who take only composition and science

3 people who take only composition and math

hope this gets you started and shows you a little how to do it

Answer 2

Draw the three overlapping circles, labeled Science, Math, and Composition. You always start with the inner most region and subtract at you go along. The seven regions will be 10(math only), 7(math and science only), 0 (science only), 3(math and composition only), 9( science and composition only), 8 (composition only) and 12 in the center (all three). The sum is 49, leaving 26 of those interviewed not taking any of the three.

Answer 3

First I would draw three intersecting circles... labeling the Math class circle "M", the science class circle "S", and the composition circle "C.

........... M ............. S

.... _________ _________

... / ............. / \ .............. \

.. / ............. / . \ .............. \

. / ............. / ... \ .............. \

./ ......... __ |____|__ ........... \

.\ ........ / ... | .... | ... \ ......... /

. \ ...... / .... \ ... / ..... \ ...... /

.. \ .... / ...... \ . / ....... \ .... /

... \__/_____. \/ _____ \ __/

........ \ ...................... /

......... \ .................... /

.......... \ .................. /

........... \__________/

.................... C

Next, I'm going to write in given info provided in the problem. The figure looks something like this ......

........... M ............. S

.... _________ _________

... / ............. / \ ............. \

.. / .. Math .. /.M\ ... Sci ... \

. / .. ONLY. / &S \ .. ONLY .\

./ ......... __ |____|__ ........... \

.\ ........ / ... |M,S,| ... \ ......... /

. \ ...... / .M. \&C/ .S.. \ ....... /

.. \ .... / .&C. \ . / .& C \ ..... /

... \__/______.\/ _____ \ __/

........ \ ...................... /

......... \ ...... Comp .... /

.......... \ .... ONLY ... /

........... \__________/

.................... C

..... M = 32 ........ S = 28

.... _________ _________

... / ............. / \ .............. \

.. / ............. / . \ .............. \

. / ....."X".... / 7. \ ...."Y"..... \ <== you get the "7" from "19-12=7"

./ ......... __ |____|__ ........... \

.\ ........ / ... | .... | ... \ ......... /

. \ ...... / .... \ 12 / ..... \ ...... / <== you get the "9" from "21-12=9"

.. \ .... / ..3.. \ . / ..9... \ .... / <== and you get the "3" from "15-12=3"

... \__/_____. \/ _____ \ __/

........ \ ...................... /

......... \ ........"Z"........ /

.......... \ .................. /

........... \__________/

............... C = 32

"X" represents the number of people who ONLY take math. To find "X", you take "the total number of students who are scheduled for math" MINUS "those who take both Math and Science" MINUS "those who take all three classes" MINUS "those who take both Math and Comp"....

"X" = (Total # of "M" students) - "M&S" - "M&S&C" - "M&C"

...... = 32 students - (19-12) - (12) - (15-12)

...... = 32 - 7 - 12 - 3

...... = 32 - 22

...... = 10 students....

So.... we just found out that 10 students are scheduled to only take math class. Replace the "X" in your diagram with "10"....

Now.... "Y" represents the number of people who ONLY take math. To find "Y", you take "the total number of students who are scheduled for science" MINUS "those who take both Math and Science" MINUS "those who take all three classes" MINUS "those who take both Science and Comp"....

"Y" = (Total # of "S" students) - "M&S" - "M&S&C" - "S&C"

...... = 28 students - (19-12) - (12) - (21-12)

...... = 28 - 7 - 12 - 9

...... = 28 - 28

...... = 0 students....

So.... we just found out that 0 students are scheduled to only take science class. Replace the "Y" in your diagram with "0"....

"Z" represents the number of people who ONLY take composition. To find "Z", you take "the total number of students who are scheduled for come" MINUS "those who take both Science and Comp" MINUS "those who take all three classes" MINUS "those who take both Math and Comp"....

"Z" = (Total # of "C" students) - "S&C" - "M&S&C" - "M&C"

...... = 32 students - (21-12) - (12) - (15-12)

...... = 32 - 9 - 12 - 3

...... = 32 - 24

...... = 8 students....

So.... we just found out that 8 students are scheduled to only take composition class. Replace the "Z" in your diagram with "8"....

So now this is what you Venn Diagram looks like.....

..... M = 32 ........ S = 28

.... _________ _________

... / ............. / \ .............. \

.. / ............. / . \ .............. \

. / ..."10".... / 7. \ ....."0"..... \

./ ......... __ |____|__ ........... \

.\ ........ / ... | .... | ... \ ......... /

. \ ...... / .... \ 12 / ..... \ ...... /

.. \ .... / ..3.. \ . / ..9... \ .... /

... \__/_____. \/ _____ \ __/

........ \ ...................... /

......... \ ........"8"........ /

.......... \ .................. /

........... \__________/

............... C = 32

Now if you add all the numbers in the circles... you get...........

# of students taking any of the 3 classes (math, science, comp)....

So.... Total = 10 + 7 + 0 + 3 + 12 + 9 + 8

................ = 49 students

The problem told you that 75 people were asked their schedules.... which mean that 75 - 49 = 26 people are taking classes which are neither of the 3 (math, science, or comp.)... meaning that math, science, and/or math are NOT to be found anywhere on their schedules....

Hope this makes sense... and hope the pictures help you visualize the problem....