In a survey of 100 students, the number s studying various languages were found to be: Spanish, 28; German, 30; French, 42; Spanish and German, 8; Spanish and French, 10; German and French, 5; all three languages, 3.
(a)How many students were studying no language?
(b)How many students had French as their only language?
(c)How many students studied German if and only if they studied French?
In a survey of 100 students, the numbers studying various languages were found to be: Spanish 28: German 30: French 42: Spanish and French 10: Spanish and German 8: German and French 5: all the three languages 3. How many students were studying no language?
a) add up all the students, then subtract from 100:
100-(28+30+42)=0
You don't need to account for the students taking multiple classes because they will already be counted in the individual class.
b) subtract the number of people taking french and another class from the total number of french students:
42-(10+5)=27
c) the number of students who took both french and German:
5
very satisfaction !
To answer these questions, we will use the principles of set theory and include the usage of Venn diagrams. Let's break it down step by step:
(a) To find the number of students who study no language, we need to find the number of students who are not studying any of the given languages.
Let's start by adding up the number of students studying each language: 28 (Spanish) + 30 (German) + 42 (French).
However, if we simply add these numbers together, we will be counting some students more than once since there are students studying multiple languages. So we need to subtract the number of students who are studying multiple languages.
Subtract the number of students studying two languages: 8 (Spanish and German) + 10 (Spanish and French) + 5 (German and French).
But wait, we have subtracted the students studying two languages twice in the previous step, so now we need to add back the number of students studying all three languages.
Add the number of students studying all three languages: 3 (Spanish, German, and French).
Now we can calculate the number of students studying no language by subtracting the total number of students studying any language from the total number of students:
100 (total students) - (28 + 30 + 42 - 8 - 10 - 5 + 3) = (100 - 46) = 54.
Therefore, there are 54 students studying no language.
(b) To find the number of students who have French as their only language, we need to subtract the number of students studying French who are also studying other languages.
Subtract the number of students studying French and another language from the total number of students studying French:
42 (French) - (10 (Spanish and French) + 5 (German and French) + 3 (Spanish, German, and French)) = (42 - 18) = 24.
Therefore, there are 24 students who have French as their only language.
(c) To find the number of students studying German if and only if they study French, we need to find the number of students who are studying both German and French, subtract the students studying German only, and those who are studying German along with other languages.
Subtract the number of students studying German only, the number of students studying French only, and the number of students studying all three languages, from the number of students studying German and French:
8 (Spanish and German) + 3 (Spanish, German, and French) - 5 (German and French) = (11 - 5) = 6.
Therefore, there are 6 students studying German if and only if they study French.