Wafula starts cycling at 8:00 am towards a certain destination at 90 km/hr. Nekesa starts cycling towards the same destination at 8:21 am at 97 km/hr. When does Nekesa catch up with Wafula?

Wafula started at 8:00 am. Nekesa started at 8:21 am.
Time difference = $8:21 \text{ am} - 8:00 \text{ am} = 21 \text{ minutes}$.
Convert minutes to hours:

$$21 \text{ minutes} = \frac{21}{60} \text{ hours} = \frac{7}{20} \text{ hours}$$

Wafula's speed = $90 \text{ km/hr}$.
Distance covered by Wafula = Speed $\times$ Time

$$D_W = 90 \text{ km/hr} \times \frac{7}{20} \text{ hr} = \frac{90 \times 7}{20} \text{ km} = \frac{9 \times 7}{2} \text{ km} = \frac{63}{2} \text{ km} = 31.5 \text{ km}$$

This is the head start distance Wafula has.

Both are moving in the same direction.
Nekesa's speed = $97 \text{ km/hr}$.
Wafula's speed = $90 \text{ km/hr}$.
Relative speed = Nekesa's speed - Wafula's speed

$$V_{relative} = 97 \text{ km/hr} - 90 \text{ km/hr} = 7 \text{ km/hr}$$

Time to catch up = Head start distance / Relative speed

$$T_{catch-up} = \frac{31.5 \text{ km}}{7 \text{ km/hr}} = 4.5 \text{ hours}$$

Convert 0.5 hours to minutes: $0.5 \times 60 = 30 \text{ minutes}$.
So, the time to catch up is 4 hours and 30 minutes.

Nekesa started at 8:21 am.
Catch-up time = Nekesa's start time + Time to catch up

$$8:21 \text{ am} + 4 \text{ hours } 30 \text{ minutes}$$ $$8:21 \text{ am} + 4 \text{ hours} = 12:21 \text{ pm}$$ $$12:21 \text{ pm} + 30 \text{ minutes} = 12:51 \text{ pm}$$

The time Nekesa caught up with Wafula is $\text{12:51 pm}$.