SOLUTION: In a survey of 100 student the number of student studying various language where find to be: Arabic-28, french-30, yoruba-42, Arabic and yoruba-10, Arabic and french-8, french and

Question 1151779: In a survey of 100 student the number of student studying various language where find to be: Arabic-28, french-30, yoruba-42, Arabic and yoruba-10, Arabic and french-8, french and yoruba-5, all the three language-3
(1). How many student where studying no language
(2). How many student had yoruba as their only language
(3). How many student study french if and only if study yoruba
Found 2 solutions by Theo, jim_thompson5910:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i think it's going to work like this.
you want to break each category up into different subgroups.
start with the students that are leaning all 3.
that's an only category, since it includes only students that are taking all 3.
then work up to the category of pairs of courses.
to make them pairs of courses only, you have to subtract the students who are taking all 3.

there are 10 students who are taking arabic and yoruba.
included is the students that are taking all 3.
subtract 3 from this category and you have 7 students who are taking arabic and yoruba only.

there are 8 students who are taking arabic and french.
included is the students that are taking all 3.
subtract 3 from this category and you have 5 students who are taking arabic and french only.

there are 5 students who are taking french and yoruba.
included is the students that are taking all 3.
subtract 3 from this category and you have 2 students who are taking french and yoruba only.

what you have now is:
28 taking arabic
30 taking french
42 taking yoruba
3 taking all 3 only.
7 taking arabic and yoruba only.
5 taking arabic and french only.
2 taking french and yoruba only.

from the 28 who are taking arabic, you have to subtract 3 taking all 3 only and 7 taking arabic and yoruba only and 5 taking arabic and french only.
this leave you with 28 - 3 - 7 - 5 = 13 who are taking arabic only.

from the 30 who are taking french, you have to subtract 3 taking all 3 only and 5 taking arabic and french only and 2 taking french and yoruba only.
this leaves you with 30 - 3 - 5 - 2 = 20 who are taking french only.

from the 42 who are taking yoruba, you have to subtract 3 taking all 3 only and
7 taking arabic and yoruba only and 2 taking french and yoruba only.
this leave you with 42 - 3 - 7 - 2 = 30 taking yoruba only.

what you have now are:

only arabic = 13
only french = 20
only yoruba = 30
only arabic and french = 5
only arabic and yoruba = 7
only french and yoruba = 2
only arabic and french and yoruba = 3

this adds up to 80 students which means 20 students aren't studying any language at all since there are 100 students total.

you can work your way back up to the original figures by adding back the sub categories into their respective categories.

for example:
13 students took arabic only.
add 5 to include the students who took arabic and french only.
add 7 to include the students who took arabic and yoruba only.
add 3 to include the students who took all 3.
you get a total of 13 + 5 + 7 + 3 = 28 students who took arabic.
5 took arabic and french only.
add 3 to include the students who took all 3.
you get a total of 5 + 3 = 8 students who took arabic and french.

a generalized formula from the original numbers would be.

number who took arabic minus number who took pairs of courses that include arabic plus number who took all 3 = number who took arabic only.
this would be 28 - 8 - 10 + 3 = 13

number who took french minus number who took pairs of courses that include french plus number who took all 3 = number who took french only.
thi would be 30 - 8 - 5 + 3 = 20

number who took yoruba minus number who took pairs of courses that include yoruba plus number who took all 3 = number who took yoruba only.
this would be 42 - 10 - 5 + 3 = 30.

working your way down, you would then subtract number who took all 3 from number who took pairs of courses.
for example, number who took arabic and french minus number who took all 3 = 7.

same idea for he others.

i made a chart which summarizes what was said above.
what it looks like is shown below.

in any case, you solution should be:
(1). How many student where studying no language
20 students were studying no language.

(2). How many student had yoruba as their only language
30 students has yoruba as their only language.

(3). How many student study french if and only if study yoruba
not sure how to interpret this one.
i think this would be the students who study french and yoruba only.
i believe this number is equal to 2.
i don't think this would include the students who studied all 3 since that would include the students who studied arabic as well.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

A = set of people who study Arabic
F = set of people who study French
Y = set of people who study Yoruba
U = universal set = set of everyone surveyed (100 people total)

Draw a Venn Diagram of 3 circles representing the three sets of language classes.

Take note of the 8 distinct regions which I've marked in blue
Region (a) = set of people taking Arabic only
Region (b) = set of people taking Arabic and French only
Region (c) = set of people taking French only
Region (d) = set of people taking Arabic and Yoruba only
Region (e) = set of people taking all three languages
Region (f) = set of people taking French and Yoruba only
Region (g) = set of people taking Yoruba only
Region (h) = set of people taking neither of the three classes mentioned

We have 3 people studying all three languages, so that goes in the very center region (e).

There are 10 people taking Arabic and Yoruba, so that leaves 10-3 = 7 who are taking Arabic and Yoruba only (and not French). This goes in region (d)

There are 8 people taking Arabic and French, leaving 8-3 = 5 people taking these two languages only (not Yoruba). This goes in region (b).

Then we have a '2' in region (f) because there are 5 people studying French and Yoruba, so 5-3 = 2 are just studying these two languages (not Arabic).

We have this partially filled out Venn Diagram so far

We know that 28 students are studying Arabic. So far we have 5, 3, and 7 in regions (b), (e) and (d). The total so far is 5+3+7 = 15. That leaves 28-15 = 13 who study Arabic only (and not French nor Yoruba). Write "13" in region (a).

Then there are 30 French students. Subtract off the values that are in circle F, and we have 30-5-3-2 = 20 students who are studying French only (neither Arabic nor Yoruba). Write this in region (c).

Next up there are 42 students in a Yoruba language class. Subtract off the values in circle Y: 42-7-3-2 = 30. Write "30" in region (g) to indicate we have 30 students who are only taking this language class (and neither French nor Arabic)

Lastly, we add all of the values we have filled out so far in regions (a) through (g)
13+5+20+7+3+2+30 = 80
There are 80 students who are taking at least one language class mentioned here (ie they are taking one or more of the three classes)

However, there were 100 people surveyed as part of the universal set. So we have 100-80 = 20 people taking neither of the three language courses. Write this in region (h)

This is the fully completed Venn Diagram

The region labels are optional, but I find them helpful to be able to describe the diagram values.

--------------------------------------------------------

Once you have the Venn Diagram, the questions are fairly straight forward

Question (1). How many student where studying no language
Answer: 20 students (see region (h))

Question (2). How many student had Yoruba as their only language
Answer: 30 students (see region (g))

Question (3). How many student study French if and only if study Yoruba
Answer: 5 students (combine regions (e) and (f))
I interpret this question to be asking "how many students are studying Yoruba and French at the same time?". This includes the 3 people taking all three languages.

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