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.
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
IN A CLASS OF 55 STUDENTS, 35 TAKE ENGLISH,40 TAKE FRENCH AND 5 TAKE OTHER LANGUAGES. DETERMINE HOW MANY STUDENTS TAKE BOTH LANGUAGES
25
To determine the number of students taking all three languages, we can start by summing up the number of students taking only French, only German, and only Spanish:
Students taking only French = 20
Students taking only German = 20
Students taking only Spanish = 14
Total students taking only one language = 20 + 20 + 14 = 54
Since we know that 35 students take French, 32 take German, and 30 take Spanish, we can calculate the number of students taking two languages by subtracting the students taking only one language from the total number of students taking each language:
Students taking French and German = 35 - 20 = 15
Students taking French and Spanish = 35 - 14 = 21
Students taking German and Spanish = 32 - 20 = 12
Now, let's address the issue of the 7 students who are taking both French and German, some of whom also take Spanish. Since the total number of students taking both French and German is 7, and this group includes some students taking Spanish, we need to consider the number of students taking all three languages.
Let's denote the number of students taking all three languages as "x".
Therefore, the number of students taking only French and German (excluding Spanish) is 7 - x.
The number of students taking French, German, and Spanish can be calculated as follows:
Students taking French, German, and Spanish = Students taking only French and German - Students taking only French, German, or Spanish
Students taking French, German, and Spanish = (7 - x) - (Total students taking only one language)
Since the total number of students taking only one language is 54, the equation becomes:
x = (7 - x) - 54
Simplifying the equation:
2x = -47
x = -47/2
However, since the number of students cannot be negative, we can conclude that there are no students taking all three languages.
To find the number of students taking none of the three languages, we need to subtract the total number of students taking any language from the total number of students in the school.
Total number of students taking any language = Students taking only French + Students taking only German + Students taking only Spanish + Students taking French and German + Students taking French and Spanish + Students taking German and Spanish
Total number of students taking any language = 20 + 20 + 14 + 15 + 21 + 12 = 102
Total number of students taking none of the three languages = Total number of students in the school - Total number of students taking any language
Total number of students taking none of the three languages = 100 - 102 = -2
Again, since the number of students cannot be negative, this means that there may be an error or inconsistency in the given information.