A bus leaves Migori at 7:00 am and travels towards Nairobi at a speed of 80 km/hr. A car leaves Migori at 8:15 am and travels towards Nairobi at a speed of 100 km/hr. Calculate: a) The time when the c

Bus:
Departure time = 7:00 am
Speed ($S_B$) = 80 km/hr

Car:
Departure time = 7:00 am + 1.25 hours = 8:15 am
Speed ($S_C$) = 100 km/hr

The car travels for $(t - 1.25)$ hours.
Distance covered by bus ($D_B$) = $S_B \times t = 80t$
Distance covered by car ($D_C$) = $S_C \times (t - 1.25) = 100(t - 1.25)$

When the car catches up, $D_B = D_C$.

$6.25 \text{ hours} = 6 \text{ hours and } 0.25 \times 60 \text{ minutes} = 6 \text{ hours and } 15 \text{ minutes}$.
Time of catch-up = 7:00 am + 6 hours 15 minutes = 1:15 pm.
The time when the car caught up with the bus is $\boxed{\text{1:15 pm}}$.

Distance = $S_B \times t$
Distance = $80 \text{ km/hr} \times 6.25 \text{ hours}$
Distance = $500 \text{ km}$.
The distance from Migori when the car caught up with the bus is $\boxed{\text{500 km}}$.

The Matatu met the bus at the same time the car caught up with the bus, which is 1:15 pm.

Bus departure time = 7:00 am
Meeting time = 1:15 pm
Time bus traveled = 1:15 pm - 7:00 am = 6 hours 15 minutes = 6.25 hours.
Distance from Migori to meeting point ($D_{B\_meet}$) = $S_B \times \text{time bus traveled} = 80 \text{ km/hr} \times 6.25 \text{ hours} = 500 \text{ km}$.

Matatu departure time = 10:10 am
Meeting time = 1:15 pm
Time Matatu traveled = 1:15 pm - 10:10 am.
1:15 pm = 13:15 (24-hour format)
10:10 am = 10:10 (24-hour format)
Time Matatu traveled = (13 hours 15 minutes) - (10 hours 10 minutes) = 3 hours 5 minutes.
Convert 5 minutes to hours: $\frac{5}{60} = \frac{1}{12}$ hours.
Time Matatu traveled = $3 + \frac{1}{12} = \frac{36+1}{12} = \frac{37}{12}$ hours.
Matatu speed ($S_M$) = 100 km/hr.
Distance from Kisumu to meeting point ($D_{M\_meet}$) = $S_M \times \text{time Matatu traveled} = 100 \text{ km/hr} \times \frac{37}{12} \text{ hours} = \frac{3700}{12} = \frac{925}{3} \text{ km}$.

Total distance = $D_{B\_meet} + D_{M\_meet} = 500 \text{ km} + \frac{925}{3} \text{ km}$
Total distance = $\frac{1500}{3} + \frac{925}{3} = \frac{2425}{3} \text{ km}$.

Time = $\frac{\text{Total distance}}{\text{Matatu speed}} = \frac{2425/3 \text{ km}}{100 \text{ km/hr}} = \frac{2425}{300} \text{ hours}$.
Simplify the fraction: $\frac{2425}{300} = \frac{97}{12}$ hours.
Convert to hours and minutes: $\frac{97}{12} = 8 \text{ with a remainder of } 1$. So, 8 hours and $\frac{1}{12}$ hours.
$\frac{1}{12} hours = \frac{1}{12}