There are a total of 103 foreign language students in a high school where they offer only Spanish, French, and German.
You are given the following details for this semester:
44 take Spanish.
44 take French.
42 take German.
7 take Spanish and French but not German.
5 take Spanish and German but not French.
5 take French and German but not Spanish.
22 students are taking at least two languages.
(1) How many students take all three languages in this semester?
(2) How many students take only French in this semester?
There are a total of 103 foreign language students in a high school where they offer only Spanish, French, and German.
Call these groups S F G SF SG FG FGS
You are given the following details for this semester:
7 take Spanish and French but not German.
SF = 7
5 take Spanish and German but not French.
SG = 5
5 take French and German but not Spanish.
GF = 5
44 take Spanish.
S + SF + SG + FGS = 44 -> S + FGS = 32
44 take French.
F + SF + FG + FGS = 44 -> F + FGS = 32
42 take German.
G + SG + FG + FGS = 42 -> G + FGS = 32
22 students are taking at least two languages.
SF + SG + GF + FGS = 22 -> FGS = 22 - 7- 5 -5 = 5
(1) How many students take all three languages in this semester? FGS = 5
(2) How many students take only French in this semester?
F + FGS = 32 -> F = 32 - 5 = 17
To solve this problem, we can use the principle of inclusion-exclusion. We will start by finding the number of students who take at least one language, and then use this information to answer the specific questions.
First, let's start by calculating the total number of students taking at least one language. To do this, we can add up the number of students taking each language, but we need to be careful not to double-count those who are taking more than one language:
Total students taking at least one language = Students taking Spanish + Students taking French + Students taking German - Students taking Spanish and French but not German - Students taking Spanish and German but not French - Students taking French and German but not Spanish + Students taking all three languages
Total students taking at least one language = 44 + 44 + 42 - 7 - 5 - 5 + Students taking all three languages
We know that 22 students are taking at least two languages, so we can substitute this value into the equation:
Total students taking at least one language = 44 + 44 + 42 - 7 - 5 - 5 + 22
Total students taking at least one language = 135 - 7 + 22
Total students taking at least one language = 150
Now, let's answer the specific questions:
(1) To find the number of students taking all three languages, we need to find the value of "Students taking all three languages." Substituting the values we know into the equation:
150 = 44 + 44 + 42 - 7 - 5 - 5 + Students taking all three languages
Simplifying this equation, we get:
Students taking all three languages = 150 - 120
Students taking all three languages = 30
Therefore, there are 30 students taking all three languages in this semester.
(2) To find the number of students taking only French, we need to subtract the number of students taking French and another language from the total number of students taking French:
Students taking only French = Students taking French - Students taking French and German but not Spanish - Students taking Spanish and French but not German + Students taking all three languages
Substituting the values we know into the equation:
Students taking only French = 44 - 5 - 7 + Students taking all three languages
Simplifying this equation, we get:
Students taking only French = 44 - 12 + Students taking all three languages
We don't know the value of Students taking all three languages yet, but we can substitute it with 30 (from the previous answer) to get:
Students taking only French = 44 - 12 + 30
Students taking only French = 62
Therefore, there are 62 students taking only French in this semester.