Salesman Speed Calculation

The problem asks for the speed required to cover the remaining distance in the remaining time.

Identify Given Information

• Total Distance: $D = 6$ km
• Total Time: $T = \frac{3}{4}$ h
• Fraction of distance covered: $\frac{1}{2}$
• Fraction of time used: $\frac{2}{3}$

Calculate First Part Details

The salesman covers half the distance in two-thirds of the total time.

• Distance covered ($D_1$): $\frac{1}{2} \times 6 \text{ km} = 3 \text{ km}$
• Time taken ($T_1$): $\frac{2}{3} \times \frac{3}{4} \text{ h} = \frac{6}{12} \text{ h} = \frac{1}{2} \text{ h}$

Calculate Remaining Distance and Time

Determine the distance and time left to cover.

• Remaining Distance ($D_{rem}$): $D - D_1 = 6 \text{ km} - 3 \text{ km} = 3 \text{ km}$
• Remaining Time ($T_{rem}$): $T - T_1 = \frac{3}{4} \text{ h} - \frac{1}{2} \text{ h} = \frac{3}{4} \text{ h} - \frac{2}{4} \text{ h} = \frac{1}{4} \text{ h}$

Calculate Required Speed

Speed is calculated as Distance divided by Time. The required speed is for the remaining part.

• Required Speed ($S_{rem}$) = $\frac{\text{Remaining Distance}}{\text{Remaining Time}}$
• $S_{rem} = \frac{3 \text{ km}}{\frac{1}{4} \text{ h}}$
• $S_{rem} = 3 \times 4 \text{ km/h}$
• $S_{rem} = 12 \text{ km/h}$

Therefore, the salesman must maintain a speed of 12 km/h for the remaining journey.