The slow guy took ##T## time. The better one did the same distance ##x## in ##0.8T##. You correctly observed that in this instance, ##x=\frac{1}{2}at^2##. We have two equivalent ##x##'s since they ran the same distance. Let's carefully label our accelerations, ##a_1## for the slow guy and ##a_2## for the fast one. Now
$$\frac{1}{2}a_1 T^2=\frac{1}{2} a_2 (0.8T)^2$$
Which easily simplifies to ##a_1=0.64a_2##. Since we want ##\frac{a_1}{a_2}##, we can just rearrange and find that ##\frac{a_1}{a_2}=0.64=16/25##.
When questions ask for the ratio of the same variable in different conditions, it's very useful to think about how that variable scales in its appropriate equation. For example, since time squared runs proportional to acceleration, this problem becomes immediate. I knew the answer was either ##16/25## or ##25/16## but I used logic to sort out which it would be by considering which person's acceleration was greater. So you need to work on thinking about your variables logically. The biggest piece of advice I can give you for physics is that you should be actively applying "physics thinking" to the algebra part. What I mean is that you can't expect to just set up equations using physics and then only use algebra. This works fine with basic problems, but as you get more advanced you will need to be following your algebra with much more than just a mathematical eye. This will motivate more complicated solutions, especially when you're struggling to find useful substitutions.
