Respuesta :
Answer:
91 people take Russian
26 people take French and Russian but not German
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the students that take French.
-The set B represents the students that take German
-The set C represents the students that take Russian.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of students that take only Franch, A \cap B is the number of students that take both French and German , A \cap C is the number of students that take both French and Russian and A \cap B \cap C is the number of students that take French, German and Russian.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There are 155 people in my school. This means that:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 155[/tex]
The problem states that:
90 take Franch, so:
[tex]A = 90[/tex]
83 take German, so:
[tex]B = 83[/tex]
22 take French, Russian, and German, so:
[tex]A \cap B \cap C = 22[/tex]
42 take French and German, so:
[tex]A \cap B = 42 - (A \cap B \cap C) = 42 - 22 = 20[/tex]
41 take German and Russian, so:
[tex]B \cap C = 41 - (A \cap B \cap C) = 41 - 22 = 19[/tex]
22 take French as their only foreign language, so:
[tex]a = 22[/tex]
Solution:
(1) How many take Russian?
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + 19 + 22[/tex]
[tex]C = c + (A \cap C) + 41[/tex]
First we need to find [tex]A \cap C[/tex], that is the number of students that take French and Russian but not German. For this, we have to go to the following equation:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]90 = 22 + 20 + (A \cap C) + 22[/tex]
[tex](A \cap C) + 64 = 90[/tex].
[tex](A \cap C) = 26[/tex]
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The number of students that take Russian is:
[tex]C = c + 26 + 41[/tex]
[tex]C = c + 67[/tex]
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Now we have to find c, that we can find in the equation that sums all the subsets:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 155[/tex]
[tex]22 + b + c + 20 + 26 + 19 + 22 = 155[/tex]
[tex]b + c + 109= 155[/tex]
[tex]b + c = 46[/tex]
For this, we have to find b, that is the number of students that take only German. Then we go to this eqaution:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]B = b + 19 + 20 + 22[/tex]
[tex]B = b + 61[/tex]
[tex]b + 61 = 83[/tex]
[tex]b = 22[/tex]
-------
[tex]b + c = 46[/tex]
[tex]c = 46 - b[/tex]
[tex]c = 24[/tex]
The number of people that take Russian is:
[tex]C = c + 67[/tex]
[tex]C = 24 + 67[/tex]
[tex]C = 91[/tex]
91 people take Russian
(2) How many take French and Russian but not German?
[tex](A \cap C) = 26[/tex]
26 people take French and Russian but not German
