LOGIC & PUZZLES / ESTIMATION
Estimate first.
Calculate second.
A rough answer with honest assumptions can be more useful than precise arithmetic on a bad model.
THE MAIN IDEA
Order of magnitude and units catch surprising mistakes.
An estimate is an approximate calculation built from simplified assumptions. It is useful before detailed arithmetic because it tells you what answer scale to expect. If a computed walking time is three seconds for a journey across town, a rough distance divided by an ordinary walking speed will expose the mistake immediately. Order of magnitude asks whether a quantity is tens, thousands, or millions before arguing over exact digits. Replace awkward values with nearby easy ones, break a big problem into parts, and label the units at every step. Units can cancel or reveal a missing conversion: multiplying kilometers by hours does not produce a speed, while dividing kilometers by hours does. Ranges make uncertainty visible. If an object may be between ten and twenty meters away and you walk between one and two meters per second, a rough time range is more honest than giving a precise single value unsupported by observation. State which input dominates the uncertainty; measuring it more carefully may improve the result far more than polishing a minor term. Fermi estimates break a large question into countable factors. To approximate daily cups of coffee in a city, estimate population, the share who drink coffee, the average cups per person, and how many days you mean. Each factor is uncertain, yet the multiplication can reveal a plausible scale and which assumptions deserve research. The purpose is not to publish an invented statistic. It is to structure the question and guide where to look for real data. Correlated assumptions can make a range too narrow: people who drink coffee may drink more than the overall population average, so population and per-person consumption are not independent. A useful estimate also includes a sanity check against a known bound or neighboring quantity. Is the answer less than the total population? Can a container hold that volume? Does the result make physical sense? Exact arithmetic cannot fix a mistaken unit, an ambiguous quantity, or a model that ignores an important factor. Estimation does not replace measurement when precision or safety matters; it points out where measurement is needed. For a live engineering system, medicine, budgeting decision, or safety calculation, use validated inputs and qualified guidance rather than relying on a classroom back-of-the-envelope answer. Write the assumptions, round deliberately, carry units, and report the level of precision the evidence actually supports. That is not laziness—it is better reasoning.
TRY THIS
Estimate a journey, then check the route.
Guess the distance to a nearby public place from a map scale and divide by a realistic walking-speed range. State your assumptions and show a time range. Then consult a current accessible route; treat real safety and traffic information as more important than the estimate.