(a) In a class of 45 students, 32 offered Physics(P), 28 offered Government(G) and 12 did not offer any of the two subjects. (i) Draw the Venn diagram to re...
Question 1 Report
(a) In a class of 45 students, 32 offered Physics(P), 28 offered Government(G) and 12 did not offer any of the two subjects. (i) Draw the Venn diagram to represent the information ; (ii) How many students offered both subjects? (iii) What is \(n(P \cup G)\)?
(b) If \(p = \frac{2u}{1 - u}\) and \(q = \frac{1 + u}{1 - u}\) ; express \(\frac{p + q}{p - q}\) in terms of u.
Answer Details
(a)
Since 12 students offered neither subject, place 12 outside the two circles. Let the number in the overlap be \(x\).
\[ (32-x)+x+(28-x)+12=45 \]
\[72-x=45\]
\[x=27\]
Thus, \(P\) only \(=32-27=5\), and \(G\) only \(=28-27=1\).
(i) The required Venn diagram is:
(ii) \[\boxed{n(P\cap G)=27}\] Therefore, 27 students offered both Physics and Government.
(iii) \[n(P\cup G)=45-12=\boxed{33}\]
(b)
\[p+q=\frac{2u}{1-u}+\frac{1+u}{1-u}=\frac{2u+1+u}{1-u}=\frac{3u+1}{1-u}\]
\[p-q=\frac{2u}{1-u}-\frac{1+u}{1-u}=\frac{2u-1-u}{1-u}=\frac{u-1}{1-u}=-1\]
Hence,
\[\frac{p+q}{p-q}=\frac{\frac{3u+1}{1-u}}{-1}=-\frac{3u+1}{1-u}=\boxed{\frac{3u+1}{u-1}}\]
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