How To Calculate The Grade Of A Slope

In elementary school mathematics, when students learn to graph simple linear functions, they are introduced to the concept of a ​slope​.

A linear function is just one with a graph represented by a straight line of some sort, with its placement and direction in relation to the ​x​- and ​y​-axes depending on the function's properties.

A linear equation has the form

\(y=mx+b\)

Where ​y​ is the dependent variable, ​m​ is the slope, and ​b​ is a quantity called the ​y​-intercept, the point the line crosses on the ​y​-axis.

But you may have also heard of a mathematical construct called a ​grade​, or a percent grade. Muddled, ambiguous terms like "slope ratio" and "grade of slope" don't help.

Are slopes and grades related? They are indeed, and both are indispensable in mathematics and engineering.

In everyday terms, a slope is a steady, sustained climb or descent. That's what it means in mathematics as well, but in a more formal way. ​The slope of a line is the change in vertical (y) distance per one-unit change in horizontal (x) distance.​

For example, if a point in a coordinate system moves 11 units in the positive ​x​-direction and four units in the negative ​y​-direction, the slope is (–4)/(11) = –0.364. The minus sign means the line angles "downhill" in relation to the horizontal ​x​-axis.

A horizontal line such as the function ​y​ = 5, in which there is no vertical change throughout, has a slope of 0. ​A vertical line,​ such as ​x​ = −3​, has an undefined slope​ as there is no horizontal change and dividing by zero is not permitted in mathematics.

The point-slope formula is helpful for determining the equation of a line when either two points or one point and the slope are known. It has the form

\(y − y_0 = m(x − x_0)\)

If you were given coordinates (12, −7) and told that the graph of the function had a slope of 1.25, you could determine the general equation:

\((y − (−7)) = 1.25(x − 12)\)
\((y + 7) = 1.25x −15\)
\(y = 1.25x − 22\)

Grade, or ​percent grade​, is just the slope expressed as a percentage. It is often used in real-life situations involving the construction of roads, the very steepest of which have surprisingly low slope values.

For example, the Pennsylvania Turnpike in the Eastern U.S. has a maximum slope of 0.03, meaning it rises or falls no more 3 feet for every 100 horizontal feet traveled over any segment. The percent grade in this instance is 100 × 0.03 = 3 percent.

In trigonometry, ​y​/​x​, or ​"rise over run,"​ is also the tangent of the angle formed by the ascending or descending line and the horizontal. This means that the inverse tangent (tan ⁻¹ or arctan on a calculator) of the slope equals this angle.

•In the grueling Tour de France, a three-week race through the mountains of Western Europe featuring the best male cyclists in the world, grades that reach 13 percent are considered extraordinarily fierce.

If you know the slope of a line, you can calculate horizontal distance traveled as a function of vertical distance, or the other way around. Say you know you're walking up a 4 percent grade. If you walk for 30 minutes and your horizontal position changes at a rate of 4 miles per hour, how much elevation have you gained?

4 mph for 30 min (1/2 hr) is 2 miles, and if the percent grade is 4, the slope is 4/100 = 0.04. Since slope is rise over run and in this case the "run" is 2 miles, the vertical gain can be found as follows:

\(\begin{aligned}
0.04 &= \frac{y}{2 \;\text{miles}}\)
\(y &= 0.04×2 \
&= 0.08 \;\text{miles, or about} \ &0.08 \;\text{mi}×5,280 \;\text{ft/mi} = 422 \;\text{ft}
\end{aligned}\)