The question you opened
How full is the card room
A card room runs 22 tables of nine seats, and roughly seven seats in ten are filled.
About how many players are in the room?
about 139
Twenty two tables of nine is 198 seats. Seven tenths of 198 lands just under 139, and rounding 198 to 200 first gives 140.
Two multiplications and one fraction. Twenty two tables of nine seats is 198 seats, which is easiest as 22 times 10 minus 22. Seven tenths of 198 is 138.6, so about 139 people are playing. The estimator's move is to round before multiplying rather than after. Call 198 two hundred, take seven tenths, and get 140. That is within one percent of the exact answer and takes about two seconds.
Since the input, roughly seven seats in ten, is itself approximate, carrying a decimal place through the calculation is false precision. Knowing which way a rounding pushes the answer is part of the skill. Rounding 198 up to 200 makes the estimate slightly high, so 140 is an upper bound and the truth is a little under it. When several roundings appear in one calculation, alternate their direction where you can, so the errors partly cancel instead of stacking.
This shape appears constantly once you look for it. Seats times occupancy gives attendance. Rooms times nights times rate gives hotel revenue. Employees times average salary gives payroll. In each case one factor is a count you can see and the other is a rate you have to guess, and the guess deserves all the attention.
A useful sanity check is to bracket the answer before computing it. The room cannot hold more than 198 people, and if it were half full it would hold about 100. Anything outside 100 to 198 is arithmetic gone wrong rather than a surprising result, and a bracket like that catches decimal point errors instantly.
Technique: Round, then take the fraction