Victor left his house riding his bike at a speed of 18 km/hr. When Victor was 1/2 km from his house, his brother Alex left the house riding his bike towards Victor at a rate of 24 km/hr. How long will it take Alex to catch up with Victor?

Answer: 5 minutes or 1/12 hour

Explanation:

Let t = time it takes for Alex to catch up with Victor. 
      s = time it took for Victor to ride 1/2 km

Note that Alex will only start catching up with Victor when he already traveled the same distance with Victor. In terms of equation:

distance traveled by Alex = distance traveled by Victor
(speed of Alex)(time of Alex) = (speed of victor)(time of victor)  (1)

Note: The amounts of time indicated in equation (1) refer to the amounts of time they spent on riding a bike when Alex catches up with Victor.

Since the speeds of Alex and Victor are given in the problem, we need to figure out the amount of time stated in equation (1).

Since Alex started to ride his bike only when Victor already traveled 1/2 km using his bike and Victor has already elapsed time s in riding 1/2 km, Alex only started to bike at time s. So Victor has elapsed a time of (t + s) riding a bike when he is caught up by Alex.  

Using equation (1),

(speed of Alex)(time of Alex) = (speed of victor)(time of victor)

24t = 18(t + s)     (2)

To solve for s, recall that we'll use the following formula

distance = (speed)(time)
[tex]\frac 1 2 = 18s \newline 18s = \frac 1 2 \newline \newline s = \frac{\frac 1 2}{18} \newline \boxed{s = \frac{1}{36}}[/tex]

Substituting this value of s to equation (1), we have 

24t = 18(t + 1/36)
24t = 18t + 18(1/36)
24t = 18t + 1/2
6t = 1/2
(6t ÷ 6) = (1/2) ÷ 6
t = 1/12 hour (Since the unit for speed in the problem is km/hr)

We can convert 1/12 hour to minutes by multiplying 1/12 by 60 since 1 hour = 60 minutes. So, t = 5 minutes. Hence, it will take 5 minutes for Alex to catch up with Victor.