Two towns A and B are 400 km apart. A motor cyclist travelling at an average speed of 60km/h left town A for town B at 1:20 pm. A matatu travelling at an average speed of 80km/h also left town A for town B at 2:00 pm. Calculate: a) the distance covered by the motor cyclist by 2:00pm, b) the distance from town A in km, where the matatu caught up with the motor cyclist, c) time when the matatu caught up with the motor cyclist.

A) Calculate:

i) The distance covered by the motor cyclist by 2:00 pm. Step 1: Determine the time the motor cyclist traveled until 2:00 pm. The motor cyclist left town A at 1:20 pm. The time elapsed until 2:00 pm is $2:00 pm - 1:20 pm = 40 \text{ minutes}$.

Step 2: Convert the time to hours. $40 minutes = \frac{40}{60} hours = \frac{2}{3} hours$

Step 3: Calculate the distance covered by the motor cyclist. Speed of motor cyclist = 60 km/h. Distance = Speed $\times$ Time $Distance = 60 km/h \times \frac{2}{3} h$ $Distance = 20 \times 2 km$ $Distance = 40 km$ The distance covered by the motor cyclist by 2:00 pm is $\boxed{40 km}$.

ii) The distance from town A in km where the matatu caught up with the motor cyclist. Step 1: Set up equations for the distance of each vehicle from town A. Let $t$ be the time in hours from 2:00 pm when the matatu catches up with the motor cyclist. At 2:00 pm, the motor cyclist has already covered 40 km (from part i). Distance of motor cyclist from A at time $t$ (after 2:00 pm): $D_{motorcyclist} = 40 + 60t$ The matatu left town A at 2:00 pm. Distance of matatu from A at time $t$ (after 2:00 pm): $D_{matatu} = 80t$

Step 2: Equate the distances to find the time $t$ when they meet. When the matatu catches up with the motor cyclist, their distances from town A are equal. $40 + 60t = 80t$

Step 3: Solve for $t$. $40 = 80t - 60t$ $40 = 20t$ $t = \frac{40}{20}$ $t = 2 hours$ This means they meet 2 hours after 2:00 pm.

Step 4: Calculate the distance from town A where they meet. Substitute $t=2$ hours into either distance equation. Using the matatu's distance: $D_{matatu} = 80t$ $D_{matatu} = 80 km/h \times 2 h$ $D_{matatu} = 160 km$ The distance from town A where the matatu caught up with the motor cyclist is $\boxed{160 km}$.

iii) Time when the matatu caught up with the motor cyclist. Step 1: Use the time $t$ calculated in part (ii). The matatu caught up with the motor cyclist 2 hours after 2:00 pm. $Meeting time = 2:00 pm + 2 hours$ $Meeting time = 4:00 pm$ The matatu caught up with the motor cyclist at $\boxed{4:00 pm}$.