# How to Calculate Percent Slope ••• winding road image by Jim Parkin from Fotolia.com
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The concept of slope is perhaps more familiar to you in everyday language than as a formal term in mathematics. In fact, they refer to the same thing: a change in vertical position accompanying a change in horizontal position. If you are moving along with no change in your elevation (i.e, vertical position with respect to some fixed reference point in a coordinate system), you might remark that there is zero slope along your path.

As is often the case in the natural sciences, a term with a general or even poetic meaning in everyday language has a very specific definition in practice. In this case, the slope of a line on a graph is its rise divided by its run, which itself may not mean anything yet. The percent slope in turn is an easy arithmetic step forward from the value of the slope itself.

## What Is the Slope in Math?

On a standard coordinate system in two dimensions, changes in horizontal (left-right) position are indicated by a change in the x-coordinate, and vertical (up-down) changes are accompanied by changes in the y-coordinate. The difference between the final and initial y-values divided by the difference between the final and initial x-values is called the slope, often designated by m.

Importantly, the sign of the changes must be preserved. This is because slopes can be positive or negative. A positive slope is associated with lines that move upward with respect to the horizontal with x-displacement, while a negative slope is associated with lines that move downward with respect to the horizontal with x-displacement.

• One common slope formula is m = (yf− yi)/(xf− xi), where the subscripts i and f denote initial and final values respectively.

## Slope Calculation Example

Example: An ant moves from the point ( −4, 5) to the point (2, −7). What is the slope of the line between them?

Applying the formula above gives

( (−7) − 5)/(2 −(−4)) = −12/6 = −2

## Slope vs. Percent Slope

Example: What is the vertical drop of a 2 percent slope in feet over a horizontal distance of 150 feet?

First, solve for the slope in decimal terms, bearing in mind that percent is just 100 times the original number:

Percent slope = 100(rise/run); −2 = 100(rise/run); (rise/run) = −2/100 = −0.02

Thus if (rise/run) = −0.02 and the "run" is 150, the "rise" in feet is actually a drop: (−0.02)(150 feet) = 3 feet.

The quotient (rise/run) has a specific meaning in trigonometry. It is the tangent of the angle between the sloping line and the horizontal (x-axis). In a right triangle, this quotient is written "(opposite side/adjacent side)" and is abbreviated tan.