To solve this problem, let's break it down step by step.

1. Total number of participants: 60

2. First half of the race:

   • Half of the participants: $\frac{60}{2} = 30$
   • Participants who drop out: $\frac{2}{3}$ of 30 $\frac{2}{3} \times 30 = 20$
   • Participants remaining after the first half: $30 - 20 = 10$

3. Second half of the race:

   • Remaining from the first half: 10
   • Participants who drop out: $\frac{2}{5}$ of 10 $\frac{2}{5} \times 10 = 4$
   • Participants remaining after the second half: $10 - 4 = 6$

Therefore, the total number of participants running the race after both halves is:

$6$

Would you like further details on any part of the solution, or do you have any additional questions?

Here are 8 related questions to expand your understanding:

1. If the total number of participants was 100, how many would be running after both halves?
2. What if $\frac{3}{4}$ of the participants dropped out in the first half instead of $\frac{2}{3}$?
3. How many participants would remain if only $\frac{1}{5}$ of the participants dropped out in the second half?
4. What percentage of the original participants finished the race?
5. If the race had three stages with equal drop rates, how many would remain?
6. What is the effect on the number of participants if the drop rate in the first half increases to $\frac{3}{4}$?
7. How many participants would remain if instead $\frac{1}{3}$ dropped out in both halves?
8. What happens if participants dropped out at different rates in different stages?

Tip: When dealing with fractions and ratios, always carefully identify which part of the whole you are calculating to avoid errors.