Readings & Meanings:
This invention was the result of researchers every day sweating from their brows.
I'm putting the certificate into a frame.
The register's balance amount was not correct, so I had to make up the difference.
Oogai is our radical this evening, 頁. It means "head." It's a pictograph of a kneeling person with a prominent head.
Our phonetic is 客, which is just an U-kammuri 宀 on top of our friend 各 (see previous posts).
各 can lend us the sense of "hard," and the forehead is the hard part of the head (for some more than others).
Amount comes from another meaning, "notice board." 額 could mean notice board, very rarely, and from a notice board you can read the amount of seating capacity. (So sayeth KN.)