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अध्ययन गर्न मन पराउदनन् ।
In a class of 100 students, 30 students like to study environment science
only and 40 students like to study economics only but 10 students don't
like to study any of these subjects.
a) माथिको सन्दर्भमा S र E ले क्रमशः वातावरण विज्ञान र अर्थशास्त्र
अध्ययन गर्न मन पराउने विद्यार्थीहरूको समूहलाई जनाउँछ भने दुबै
विषयमध्ये कुनै पनि विषय मन नपराउने भन्ने विद्यार्थीको समूहलाई
गणनात्मकता सङ्केतमा लेख्नुहोस् ।
In the above context, S and E denote the sets of students who like
to study environmental science and economics respectively. Write
the cardinal notation of the set of students who don't like any of
these subjects.
माथिको तथ्यलाई भेनचित्रमा प्रस्तुत गर्नुहोस् ।
b)
Present the above information in a Venn-diagram.
(1)
(1)
अर्थशास्त्र अध्ययन गर्न मन पराउने विद्यार्थीहरुको संख्या पत्ता लगाउनुहोस् ।
Find the number of students who like to study economics.
d) के n(E) = जम्मा विद्यार्थीको 30% हुन्छ ? कारण दिनुहोस् ।
Is n(E) = 30% of the total students? Give reason.
(3)
(1)
ce
​

Sagot :

dashataran

Answer:

[tex]\noindent\rule{12cm}{0.4pt}[/tex]

a) Cardinal Notation of the Set of Students who don't like any of these subjects:

  • We know that there are 10 students who don't like to study any of the subjects.

    [tex]\fontsize{12}\selectfont\text{$\bullet $\ Let $U$ be the universal set representing all students in the}[/tex]

       [tex]\fontsize{12}\selectfont\text{class, so $n(U)=100.$}[/tex]

    [tex]\fontsize{12}\selectfont\text{$\bullet $\ Let $S$ be the set of students who like to study environmental}[/tex]

       [tex]\fontsize{12}\selectfont\text{science.$}[/tex]

    [tex]\fontsize{12}\selectfont\text{$\bullet$\ Let $E$ be the set of students who like to study economics. }[/tex]

    [tex]\fontsize{12}\selectfont\text{$\bullet$\ So, the set of students who don't like any of these subjects is}[/tex]

        [tex]\fontsize{12}\selectfont\text{denoted by $(SUE)'.$}[/tex]

   Thus the cardinal notation of this set is:

        [tex]\fontsize{12}\selectfont\text{$n(SUE)'.$}[/tex]

[tex]\noindent\rule{12cm}{0.4pt}[/tex]

b)  Please have a look at the image I posted for the Venn diagram.

[tex]\noindent\rule{12cm}{0.4pt}[/tex]

c)  Find the number of students who like to study economics.        

    [tex]\fontsize{12}\selectfont\text{$\bullet$\ Total number of students, $n(U)=100.$}[/tex]

    [tex]\fontsize{12}\selectfont\text{$\bullet$ Students who like only environmental science, $n_o(S)=30$}[/tex]

    [tex]\fontsize{12}\selectfont\text{$\bullet$\ Students who like only economics, $n_o(E)=40.$}[/tex]

    [tex]\fontsize{12}\selectfont\text{$\bullet$\ Students who don't like any subjects, $n(SUE)'=10$}[/tex]

Using the formula:

    [tex]\fontsize{12}\selectfont\text{$n(U)=n_o(S)+n_o(E)+n(S\cap E)+n(SUE)'$}[/tex]

Substitute the known values:

             [tex]\fontsize{12}\selectfont\text{$100=30+40+n(S\cap E)+10$}[/tex]

Solve for [tex]\fontsize{12}\selectfont\text{$n(S\cap E):$}[/tex]

             [tex]\fontsize{12}\selectfont\text{$100=80+n(S\cap E)$}[/tex]

             [tex]\fontsize{12}\selectfont\text{$n(S\cap E)=20$}[/tex]

So, 20 students like both environmental science and economics.

To find the number of students who like to study economics, we consider:

              [tex]\fontsize{12}\selectfont\text{$n(E)=n_o(E)+n(S\cap E)$}[/tex]

Substitute the known values:

              [tex]\fontsize{12}\selectfont\text{$n(E)=40+20=60$}[/tex]    

[tex]\noindent\rule{12cm}{0.4pt}[/tex]

d) Is n(E) = 30% of the total students? Give reason.

  • We found that the number of students who like to study economics is 60.
  • The total number of students in the class is 100.

[tex]\fontsize{12}\selectfont\text{To find if $n(E)$ is 30\% of the total students, we calculate:}[/tex]

           [tex]\fontsize{12}\selectfont\text{$\dfrac{n(E)}{n(U)}\times100\%.$}[/tex]      

[tex]\fontsize{12}\selectfont\text{Substituting the values, we get:$}[/tex]

          [tex]\fontsize{12}\selectfont\text{$\dfrac{60}{100}\times100\%=60\%$}[/tex]

Therefore, n(E) is not 30% of the total students.[tex]\noindent\rule{12cm}{0.4pt}[/tex]

View image dashataran
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