If the length of the first road is x then the length of the second is x-10. If time taken for the first journey is t, t, is distance divided by speed=x/50. The time taken for the second journey is (x-10)/55=t-1/3. If we substitute t=x/50 in this equation we get (x-10)/55=x/50-1/3. First, multiply through by 3: 3(x-10)/55=3x/50-1. Now multiply through by 50: 150(x-10)/55=3x-50. Multiply through by 55: 150(x-10)=165x-2750; 150x-1500=165x-2750; 15x=1250, so x=1250/15=250/3=83.33 miles. Therefore x-10=73.33 miles.
Let's check the result. The first journey takes 83.33/50=1.667 hours; the second journey takes 73.33/55=1.333 hours. The difference in time is 0.33 hours = 20 mins. The longer route is 83.33 miles and the shorter route is 73.33 miles.