(a)(i) \(p > q .... (1)\)
\(p + q = 3(p - q) ...... (2)\)
From (2), \(p + q = 3p - 3q\)
\(p - 3p = - 3q - q \implies - 2p = - 4q\)
\(p = 2q\)
(ii) \(\frac{p^{2} + q^{2}}{pq}\)
= \(\frac{(2q)^{2} + q^{2}}{2q \times q}\)
\(\frac{4q^{2} + q^{2}}{2q^{2}}\)
= \(\frac{5q^{2}}{2q^{2}}\)
= \(\frac{5}{2}\)
(b) Selling price = \(\frac{100 \times 66}{25}\)
= \(N264.00\)
Using \(\frac{SP - CP}{CP} = % gain\)
\(\frac{264 - CP}{CP} = \frac{32}{100}\)
\(100(264 - CP) = 32CP\)
\(26400 = 32CP + 100CP = 132CP\)
\(CP = \frac{26400}{132}\)
= \(N200\)
When he sells 20 for N50, Selling price = \(\frac{100 \times 50}{20} = N250.00\)
Hence, he made gain.
\(% gain = \frac{250 - 200}{200} \times 100%\)
\(\frac{1}{4} \times 100%\)
= \(25% gain\)