Let us try to make it as simple as possible. If we think in the opposite direction, that Derek has traveled 178 miles, then traveled 50 miles, how much has he traveled? Well, if in the question the first part would be 1/3, then the answer to my question would be 2/3, right? This means that 2/3 is equivalent to those 50 miles, and 178 miles he has traveled. But 2/3 of what? Of the total distance, right? Let us call the total distance L, to make things easier.
Ok now let us express the English words into Mathematics word (I spell it with a capital 'M' just to denote that Mathematics is actually a language!)
[tex]L \cdot \frac{2}{3}=50 \text{ miles } +178 \text{ miles }[/tex]
Great! Now it is just a matter of calculating everything and solving for L. Having L, we have an answer. So let us do this:
[tex]L \cdot \frac{2}{3}=50 \text{ miles } +178 \text{ miles } \Rightarrow L \cdot \frac{2}{3}=228 \text{ miles } \Rightarrow L = \frac{3}{2}\cdot 228 \text{ miles }[/tex]
[tex]L = 342 \text{ miles.}[/tex]
Voila!
