A grade on its own is a shape. Attach it to a known elevation and it becomes a buildable line: $E_2 = E_1 + \dfrac{G L}{100}$, read aloud E-two equals E-one plus G L over one hundred.

Name every letter, and fix the subscripts once. $E_1$ is the elevation you know — the benchmark, the manhole rim, the invert already surveyed — in metres above datum. $E_2$ is the elevation you want, at the far end, in the same metres above the same datum. Subscript 1 is always where you start and 2 is always where you are going, so if you reverse the direction of travel you must reverse the subscripts too. $L$ is the horizontal run between them in metres, and $G$ is the percent grade. You are solving for $E_2$ going down the line, for $G$ when an as-built has handed you both ends, and for $L$ when you need to know where a target elevation is reached.

Read the shape of it. $\dfrac{G L}{100}$ is nothing but the rise from the last lesson — the grade spends itself over the run — and $E_1$ is what ties the whole thing to datum. Drop $E_1$ and you have computed a change, not an elevation, and a change cannot be set out.

Here is the discipline that separates a clean cut sheet from a rebuilt one. The sign lives in $G$, and nowhere else. A falling line is a negative grade, so you carry $-2.00$ into the formula and let the addition handle the direction. Never rewrite the formula with a minus for a downgrade — do both and the two negatives cancel, and your invert climbs. On a gravity sewer that error puts the downstream invert above the upstream one, and the pipe is in the ground before anybody notices the water will not go.

One habit worth building: after every projection, ask whether the answer moved the way the drawing said. Falling grade, smaller number. Rising grade, larger number. It takes a second and it catches the most expensive mistake in this chapter.