(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 mak...

Answer Details

(a) Solve \(\dfrac{3}{2p-\tfrac{1}{2}} = \dfrac{\tfrac{1}{3}}{\tfrac{1}{4}p + 1}\)

Cross multiply:

\[3\left(\tfrac{1}{4}p + 1\right) = \tfrac{1}{3}\left(2p - \tfrac{1}{2}\right)\]

Expand both sides:

\[\tfrac{3}{4}p + 3 = \tfrac{2}{3}p - \tfrac{1}{6}\]

Collect the \(p\) terms and constants:

\[\tfrac{3}{4}p - \tfrac{2}{3}p = -\tfrac{1}{6} - 3\]

\[\left(\tfrac{9-8}{12}\right)p = -\tfrac{19}{6} \;\Rightarrow\; \tfrac{1}{12}p = -\tfrac{19}{6}\]

\[p = -\tfrac{19}{6}\times 12 = -38\]

\(p = -38\).

(b) Actual percentage profit

The marked price of GH¢760.00 gives a 20% profit, so it is \(120\%\) of the cost price:

\[\text{Cost price} = \frac{760}{1.20} = \text{GH¢}633.33\]

Sold at a 5% discount off the marked price:

\[\text{Selling price} = 760\times 0.95 = \text{GH¢}722.00\]

Actual profit \(= 722.00 - 633.33 = \text{GH¢}88.67\).

\[\text{Percentage profit} = \frac{88.67}{633.33}\times 100 \approx 14\%\]

Actual percentage profit \(\approx 14\%\) (to 2 s.f.).