The question you opened
Fifteen percent up front
An invoice comes to 2,400 dollars and the deposit is 15 percent.
How large is the deposit, in dollars?
$360
Ten percent of 2,400 is 240, and half of that is 120. Fifteen percent is the two added together, so 360 dollars.
Ten percent of 2,400 is 240, found by moving the decimal point one place. Half of 240 is 120, which is 5 percent. Adding them gives 15 percent, so the deposit is 360 dollars and the balance is 2,040. Almost every percentage anyone actually uses can be built from three pieces: a tenth, a half of a tenth, and a hundredth. Twenty percent is two tenths.
Twenty five percent is a quarter, so halve twice. Thirty five percent is three tenths plus half a tenth. Seven percent is seven hundredths, or a tenth minus three hundredths. Nothing needs multiplication by a decimal. The order matters for accuracy. Take the tenth first, while the number is still whole, then halve. Halving 2,400 to 1,200 and then trying to reach 15 percent from there means working with 12.5 percent of something, which is where mistakes appear.
Percentages are also reversible, and the reverse direction is where people lose money. If 360 is the 15 percent deposit, the total is 360 divided by 0.15, which is 360 times 20 over 3, so 2,400. More usefully: if a price after a 20 percent discount is 240, the original was 240 divided by 0.8, which is 300 rather than 288.
Dividing by the fraction that survived is the only correct move, and adding the discount back is the common error. For a deposit specifically, the number worth checking is the balance rather than the deposit. Knowing that 2,040 is still due is what makes the 360 meaningful, and that subtraction takes one more second.
Technique: Ten percent plus half of it