(a) If \(\frac{3}{2p – \frac{1}{2}} = \frac{\frac{1}{3}}{\frac{1}{4}p + 1}\), find p.
(b) A television set was marked for sale at GH¢ 760.00 in order to make a profit of 20%. The television set was actually sold at a discount of 5%. Calculate, correct to 2 significant figures, the actual percentage profit.
Explanation
(a) \(\frac{3}{2p - \frac{1}{2}} = \frac{\frac{1}{3}}{\frac{1}{4}p + 1}\)
\(\implies 3(\frac{1}{4}p + 1) = \frac{1}{3}(2p - \frac{1}{2})\)
\(\frac{3}{4}p + 3 = \frac{2}{3}p - \frac{1}{6}\)
\(\frac{3}{4}p - \frac{2}{3}p = - 3 - \frac{1}{6}\)
\(\frac{1}{12}p = - 3\frac{1}{6}\)
\(p = \frac{-19}{6} \div \frac{1}{12}\)
\(p = \frac{-19}{6} \times 12 = -38\)
(b) \(5% \times ¢760.00 = \frac{5}{100} \times ¢760 = ¢38.00\)
So the TV set was actually sold for ¢(760 - 38) = ¢722.00
Using,
\(% profit = \frac{\text{SP - CP}}{CP} \times 100%\)
\(20% = \frac{760 - x}{x} \times 100%\)
\(20% x = (760 - x) \times 100%\)
\(x = 5(760 - x) \implies x = 3800 - 5x\)
\(6x = 3800 \implies x = ¢633.33\)
Thus, the actual profit = actual SP - CP
= ¢(722 - 633.33)
= ¢88.67
Hence, actual %age profit = \(\frac{88.67}{633.33} \times 100%\)
= 14.0006%
= 14% (2 sig. figs)