(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.
Explanation
(a) (i) 
(ii) \(32 - x + x + 28 - x + 12 = 45\)
\(72 - x = 45\)
\(x = 72 - 45 = 27\)
27 students offered both Physics and Government.
(iii) \(n(P \cup G) = 45 - 12 = 33\)
(b) \(p = \frac{2u}{1 - u} ; q = \frac{1 + u}{1 - u}\)
\(\frac{p + q}{p - q}\)
\(p + q = \frac{2u}{1 - u} + \frac{1 + u}{1 - u}\)
= \(\frac{2u + 1 + u}{1 - u}\)
= \(3u + 1}{1 - u}\)
\(p - q = \frac{2u - 1 - u}{1 - u}\)
= \(\frac{u - 1}{1 - u}\)
= \(\frac{-(1 - u)}{1 - u}\)
= \(-1\)
\(\frac{p + q}{p - q} = \frac{\frac{3u + 1}{1 - u}}{-1}\)
= \(\frac{-(3u + 1)}{1 - u}\)
= \(\frac{-(3u + 1)}{-(u - 1)}\)
= \(\frac{3u + 1}{u - 1}\)