You're on a roll — here are the solutions: 5. In the figure below angles QPR and PSR are equal. Angle PRS is a right angle. The ratio of PR:QR = 4:3. Given that the area of triangle PQR is $24 \text{ cm}^2$, find the area of triangle PRS. Step 1: Identify similar triangles. Given that $\angle PRS = 90^\circ$, and from the figure, $\angle PRQ = 90^\circ$. Given $\angle QPR = \angle PSR$. Since the sum of angles in a triangle is $180^\circ$: In $\triangle PQR$, $\angle PQR = 180^\circ - 90^\circ - \angle QPR = 90^\circ - \angle QPR$. In $\triangle PRS$, $\angle RPS = 180^\circ - 90^\circ - \angle PSR = 90^\circ - \angle PSR$. Since $\angle QPR = \angle PSR$, it follows that $\angle PQR = \angle RPS$. Therefore, $\triangle PQR$ is similar to $\triangle SPR$ (Angle-Angle-Angle similarity). Step 2: Use the given area of $\triangle PQR$ and the ratio PR:QR to find the lengths of PR and QR. Let $PR = 4k$ and $QR = 3k$ for some constant $k$. The area of $\triangle PQR = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times QR \times PR$. $$ 24 = \frac{1}{2} \times (3k) \times (4k) $$ $$ 24 = \frac{1}{2} \times 12k^2 $$ $$ 24 = 6k^2 $$ $$ k^2 = 4 $$ $$ k = 2 $$ So, $PR = 4 \times 2 = 8 \text{ cm}$ and $QR = 3 \times 2 = 6 \text{ cm}$. Step 3: Use the property of similar triangles that the ratio of their areas is equal to the square of the ratio of their corresponding sides. Since $\triangle PQR \sim \triangle SPR$, the ratio of their areas is: $$ \frac{\text{Area}(\triangle PQR)}{\text{Area}(\triangle SPR)} = \left(\frac{QR}{PR}\right)^2 $$ Using the sides $QR=6$ and $PR=8$: $$ \frac{24}{\text{Area}(\triangle SPR)} = \left(\frac{6}{8}\right)^2 $$ $$ \frac{24}{\text{Area}(\triangle SPR)} = \left(\frac{3}{4}\right)^2 $$ $$ \frac{24}{\text{Area}(\triangle SPR)} = \frac{9}{16} $$ Step 4: Solve for the area of $\triangle PRS$. $$ \text{Area}(\triangle SPR) = 24 \times \frac{16}{9} $$ $$ \text{Area}(\triangle SPR) = \frac{8 \times 3 \times 16}{3 \times 3} $$ $$ \text{Area}(\triangle SPR) = \frac{8 \times 16}{3} $$ $$ \text{Area}(\triangle SPR) = \frac{128}{3} $$ $$ \text{Area}(\triangle SPR) = 42 \frac{2}{3} \text{ cm}^2 $$ The area of triangle PRS is $\boxed{42 \frac{2}{3} \text{ cm}^2}$. 6. A suit whose marked price is sh. 8,000 is sold to customer after allowing him a discount of 13%. If the trader makes a profit of 20%, find how much the trader paid for the suit. Step 1: Calculate the selling price (SP) after the discount. Marked Price (MP) = Sh. 8,000. Discount = 13%. Discount amount = $13\%$ of $8,000 = 0.13 \times 8,000 = \text{Sh. } 1,040$. Selling Price (SP) = MP - Discount amount = $8,000 - 1,040 = \text{Sh. } 6,960$. Alternatively, $SP = MP \times (1 - \text{discount percentage}) = 8,000 \times (1 - 0.13) = 8,000 \times 0.87 = \text{Sh. } 6,960$. Step 2: Calculate the cost price (CP) using the profit percentage. Profit = 20%. The selling price (SP) is the cost price plus the profit. $SP = CP + (20\% \text{ of } CP) = CP \times (1 + 0.20) = 1.20 \times CP$. $$ 6,960 = 1.20 \times CP $$ $$ CP = \frac{6,960}{1.20} $$ $$ CP = \frac{6,960}{\frac{120}{100}} = \frac{6,960 \times 100}{120} $$ $$ CP = \frac{696,000}{120} = \frac{69,600}{12} $$ $$ CP = 5,800 $$ The trader paid $\boxed{\text{Sh. } 5,800}$ for the suit. **7. Evans is a salesperson with Ajab Millers. He is paid on monthly basis as the agreement; basic pay of Ksh. 20,000, a commission