Question

10. In an examination, 35% students failed in Science, 42% failed in Social Studies failed in both the subjects. Find the: (1) percentage of total failed students. (ii) percentage of total passed students. (iii) total number of students, if 165 passed in both the subjects.

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Answer 1

$\\ \huge{\underline{\underline{\sf{\pmb{\purple{Appropriate \:Question:}}}}}}$

In an examination, 35% students failed in Science, 42% failed in Social Studies and 32% failed in both the subjects. Find the:

• (1) percentage of total failed students.
• (ii) percentage of total passed students.
• (iii) total number of students, if 165 passed in both the subjects.

$\\ \huge{\underline{\underline{\sf{\pmb{\purple{Required\: Answer:}}}}}}$

$\\ \large{\underline{\underline{\sf{\frak{Given:}}}}}$

• 35% students failed in science.
• 42% failed in social studies.
• 32% failed in both subjects.

$\\ \large{\underline{\underline{\sf{\frak{To \: Find:}}}}}$

• percentage of total failed students.
• percentage of total passed students.
• total number of students, if 165 passed in both the subjects.

$\\ \large{\underline{\underline{\sf{\frak{Solution:}}}}}$

Let,

• $\sf{Students \: failed \: in \: science = \bold{n(Sc) = 35}}$

• $\sf{Students \: failed \: in \: science = \bold{n(Soc) = 42}}$

• $\sf{Students \: failed \: in \: both \: subjects= \bold{n(Sc \cap Soc) = 32}}$

• $\sf{Students \: failed \: in \: total = \bold{n(Sc \cup Soc)}}$

According to the formula of sets:-

$\\ \underline{\boxed{\sf{n{(A \cup B)} = n(A) + n(B) - n(A \cap B)}}}$

$\\ \sf{\therefore{n(Sc \cup Soc) = n(Sc) + n(Soc) - n(Sc \cap Soc)}}$

$\\ \sf{\implies{n(Sc \cup Soc) = (35 + 42 - 32)\%}}$

$\\ \sf{\implies{n(Sc \cup Soc) = \red{45\%}}}$

$\\ \underline{\rm{Hence, the \: percentage \: of \: the \: failed \: student \: is \: \green{\bold{45\%}}}}$

Again,

• Let the percentage of total students be = n(U) = 100%

$\\ \sf{\therefore{Percentage \: of \: the \: students \: passed \: in \: exam =}}$

$\sf{\implies{n(U) - n(Sc \cup Soc) = (100 - 45)\% = \red{55\%}}}$

$\\ \underline{\rm{Hence, the \: percentage \: of \: the \: passed \: student \: is \: \green{\bold{55\%}}}}$

Now,

• Let the total number of students be x
• Percentage of the students who passed in exam = 55%

According to the question:-

$\\ \bold{55\% \: of \: x = 165}$

$\\ \sf{\implies{\dfrac{55}{100} \times x = 165}}$

$\\ \sf{\implies{\dfrac{55x}{100} = 165}}$

$\\ \sf{\implies{55x = 16500}}$

$\\ \sf{\implies{x = \dfrac{16500}{55}}}$

$\\ \sf{\therefore{x = \red{300}}}$

$\\ \underline{\rm{Hence, the \: total \: number \: of \: the \: students \: is \: \green{\bold{300}}}}$

$\\ \huge{\underline{\underline{\sf{\pmb{\purple{Final \: Answers:}}}}}}$

• Percentage of total failed students = 45%
• Percentage of total passed students = 55%
• Total number of students = 300