Question
At a school with 100 students, 35 take French, 32 take German, and 30 take Spanish. Twenty students take only French, 20 take only German, and 14 take only Spanish. In addition, 7 students are taking both French and German, some of whom also take Spanish. How many students are taking all 3 languages? How many are taking none of these 3 languages?
I have drawn a Venn Diagram with 3 circles and put in the 20, 20, and 14, but I don't know what to do with the 7 since some of the 7 is also in the intersection of all 3. When I know that, I think I can solve for the missing regions in the Venn diagram.
I have drawn a Venn Diagram with 3 circles and put in the 20, 20, and 14, but I don't know what to do with the 7 since some of the 7 is also in the intersection of all 3. When I know that, I think I can solve for the missing regions in the Venn diagram.
Answers
Reiny
We don't know how many take all three, which is our first main question.
So in your Venn diagram, place x in the intersection of all three circles.
Now look at the intersection of the F and G circles
We are told that this is 7 but x are already counted, ("some of whom also take Spanish")
So place 7-x in the region of ONLY F and G
Now look at the F circle , so far we have
20 + x + 7-x = 27
We are told that 35 take F
So place 8 in the region ONLY F and S
Do the same for circle G, placing 5 in the G and S only circle.
We can now find x from the S circle
14+8+5+x = 30
x = 3
So 3 students study all three languages
remember that some students don't take any of the three languages.
Check:
Sum of all entries in our circles = 74
So 100-74 or 26 don't take any language course
So in your Venn diagram, place x in the intersection of all three circles.
Now look at the intersection of the F and G circles
We are told that this is 7 but x are already counted, ("some of whom also take Spanish")
So place 7-x in the region of ONLY F and G
Now look at the F circle , so far we have
20 + x + 7-x = 27
We are told that 35 take F
So place 8 in the region ONLY F and S
Do the same for circle G, placing 5 in the G and S only circle.
We can now find x from the S circle
14+8+5+x = 30
x = 3
So 3 students study all three languages
remember that some students don't take any of the three languages.
Check:
Sum of all entries in our circles = 74
So 100-74 or 26 don't take any language course
LYDIA
IN A CLASS OF 55 STUDENTS, 35 TAKE ENGLISH,40 TAKE FRENCH AND 5 TAKE OTHER LANGUAGES. DETERMINE HOW MANY STUDENTS TAKE BOTH LANGUAGES
Liv
25