The question you opened
Turnout in a city runoff
A city has 900,000 people old enough to vote, and about one in eight turns out for a runoff.
About how many people vote?
about 112,500
One in eight is an eighth, and an eighth is three halvings. Nine hundred thousand halves to 450,000, then 225,000, then 112,500.
The estimate is an eighth of 900,000. Dividing by 8 is dividing by 2 three times, and halving is the one operation almost nobody gets wrong. Nine hundred thousand becomes 450,000, then 225,000, then 112,500. No long division, no calculator, and the answer is exact rather than approximate. Halving repeatedly covers more ground than it looks. It handles division by 2, 4, 8, 16, and 32 directly.
Combined with a tenth it handles 5, 20, 40, and 50, since dividing by 5 is halving and then moving the point, and dividing by 20 is halving twice and moving the point. Most everyday divisors fall into that set. Estimates like this one are graded by range rather than by exactness, which is the honest way to score them.
A guess of 100,000 is within 12 percent and is a good estimate. A guess of 900,000 divided by 8 done sloppily as 90,000 is within 20 percent and still usable for deciding how many polling stations to open. What matters is whether the answer is the right order of magnitude and close enough to act on.
The wider habit is to replace awkward fractions with chains of halves and tenths. A third is harder, so approximate it as a tenth times three and a bit, or divide by 3 directly if the number is friendly. A seventh is close to 14 percent, which is a tenth plus a twentieth plus a little.
Turnout, tips, discounts, and unit prices all live in this territory, and the arithmetic is the same in each. A last note on the scenario itself: local runoffs really do draw far fewer voters than headline elections, which is why an eighth is a plausible figure rather than a pessimistic one.
Technique: Halve it three times