In the problem:
A straight traail with a uniform inclination leads from a hotel, elevation $5000$ feet, to take a lake in a valley, elevation $600$ feet. The Length of the traill is $4100$ feet. What is the inclination (grade) of the trail?
I solved it like this: $$grade = \frac{rise}{run}$$
$$\frac{5000-600}{4100}$$
$$\frac{4400}{4100}$$
$$\text{grade}=\frac{44}{41} \ \text{or} \ 1.07317...$$
In other reference, it says that it's needed to be in percent, but what I know is that it's okay to be in just decimal or fraction. Is what I know right?
How do you visualize this? I can't think of how it can be drawn like a right triangle.
Your calculations are correct. The way you can describe the grade is either as a decimal ($1.07$), as a fraction (in your case, $1$ in $0.93$), or a percentage ($107\%$). It depends on what the quiz is asking (e.g. "Give your answer as a percentage grade").