In a survey conducted at a University, it was found that 51% of the students wanted to learn French as a foreign language, 48% wanted to learn German and 52% wanted to learn Russian. Of the surveyed students, 21% wanted to learn both French and German, 23% wanted German and Russian and 24% wanted French and Russian. Only 12% wanted to learn all three languages. A total of 500 students were surveyed.

Questions:
(1)How many students wanted to learn only German?
(2)What is the percentage of students interested in French and Russian only?
(3)How many more students (apart from those who wanted to learn French and Russian only) were interested in either French or Russian?
(4)How many students were not interested in any of the languages?
(5)What is the ratio of the number of students interested in exactly two languages to those interested in only one language?
Solution:
To solve this question set, it is absolutely critical to mark the different sections of the Venn diagram appropriately or you may stand to get confused.

Let F denote French, G denote German and R denote Russian.

Total number of students surveyed = n(U) = 100% = 500

So, it follows that n(F) = 51%, n(G) = 48% and n(R) = 52%
To represent the fact that there are students who wanted to learn more than one language, we should use the intersection notation as it represents elements common to two or more sets. As per the information given in the question statement, we have:

n(F G) = 21%, n(G R) = 23%, n(R F)
= 24%, n(F G R) = 12%



As per the diagram, it is seen that:
The percentage of students who wanted to learn F and G only = the percentage of students who wanted to learn both F and G – the percentage of students who wanted to learn all 3 languages

= n(F G) n(F G R)
= (21 12)%
= 9%

Similarly,
The percentage of students who wanted to learn G and R only
= n(G R) n(F G R)
= (23 12) %
= 11%

The percentage of students who wanted to learn R and F only
= n(R F) n(F G R)
= 12%

Now that we have all this information, we can go ahead and answer the questions.

(1)The percentage of students who wanted to learn G only = The percentage of students who wanted to learn G the percentage of students who wanted to learn G and R only the percentage of students who wanted to learn F and G only the percentage of students who wanted to learn all 3 languages = (48 11 9 12)% = 16%
Therefore, the number students who wanted to learn German only = 0.16 500 = 80

(2)The percentage of students who wanted to learn R and F only = 12%
Therefore, the number of students who wanted to learn R and F only
= 0.12 500 = 60

(3)The percentage of students who wanted either R or F = n(R F)
= n(R) + n(F) n(R F)
= (52 + 51 24)%
= 79%

The number of students who wanted either R or F = 0.79 500 = 395
Therefore, 395 60 = 335 more students were interested in either French or Russian.

(4)The percentage of students not interested in any of the languages = 100% (the percentage of students interested in any one language only) (the percentage of students interested in any two languages only) (the percentage of students interested in all three languages)
Percentage of students interested in F only = 51 9 12 12 = 18%
Similarly we can find that the percentage of students interested in only Russian and German is 17% and 16% respectively.

The percentage of students not interested in any of the languages
= 100% (18 + 17 + 16) % (9 + 12 + 11)% 12% = 5%
Therefore, the number of students not interested in any language = 0.05 500 = 25

(5)The required ratio is given by: