jmc

It is well known that $\text{Distance}=\text{Speed}\cdot \text{Time}$. In the question, we want distance. From the question, we have that the time is $60$ minutes or $1$ hour. By the equation derived from $\text{Distance}=\text{Speed}\cdot \text{Time}$, we have $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$, so the speed is $1$ mile per $x$ minutes. Because we want the distance, we multiply the time and speed together yielding $60\text{ mins}\cdot \frac{1\text{ mile}}{x\text{ mins}}$. The minutes cancel out, so now we have $\frac{60}{x}$ as our distance for the first day. The distance for the following days are: $$\frac{60}{x},\frac{60}{x+5},\frac{60}{x+10},\frac{60}{x+15}.$$ We know that $x,x+5,x+10,x+15$ are all factors of $60$, therefore, $x=5$ because the factors have to be in an arithmetic sequence with the common difference being $5$ and $x=5$ is the only solution. $$\frac{60}{5}+\frac{60}{10}+\frac{60}{15}+\frac{60}{20}=12+6+4+3=\boxed{25}.$$