Quick Estimation: Rounding, Benchmarks and Sanity Checks

Round so the errors cancel

Round each number to something easy, but round some up and some down. If you always round the same way, the mistakes pile up in one direction. (For the formal rules on rounding, see rounding numbers: rules, ties and errors.)

Worked example: a shopping basket holds items at 3.99, 12.49, 7.95, 2.10 and 5.99. Rounding to friendly values gives 4 + 12.50 + 8 + 2 + 6 = 32.50. The exact total is 32.52, so the estimate is off by two cents. If you had rounded every price up to the next whole number, you would get 4 + 13 + 8 + 3 + 6 = 34, which is about 4.6% too high.

Multiplication behaves differently, because both rounding choices affect the result. For 48 × 19, rounding both numbers up to 50 × 20 gives 1,000. The exact answer is 912, so the estimate is about 10% high. A more careful route is 48 × 20 = 960, then subtract one 48 to reach 912. Either way, you know the answer is near 1,000 and below it, so a calculator showing 9,120 or 91.2 must contain a typing slip.

When both factors are rounded up, the estimate is an upper bound. When both are rounded down, it is a lower bound. That is useful when you only need to know whether something is safely under or over a limit.

Benchmarks worth remembering

A few anchor values let you estimate conversions and sizes without any tool. The last column shows how close each shortcut is.

An inch is exactly 2.54 cm, and a liter of water weighs about a kilogram. Those two cover a surprising number of everyday conversions. For more shortcuts and the error each one carries, see mental shortcuts for metric and imperial conversions.

Build a ballpark from assumptions

When a number is unknown, break it into pieces you can guess and multiply them. These are called Fermi estimates. The skill is in stating each assumption openly, so anyone can swap in a better value.

Worked example, yearly commuting fuel cost. Assume a 25-mile round trip each working day, 250 working days a year, a car doing 25 miles per gallon, and fuel at 3.50 per gallon. Distance is 25 × 250 = 6,250 miles. Fuel is 6,250 ÷ 25 = 250 gallons. Cost is 250 × 3.50 = 875 per year.

The point is not the exact figure but how sensitive it is. If the car does 30 mpg instead, fuel drops to 208 gallons and the cost to about 729. If the round trip is 20 miles instead of 25, distance and cost fall by 20%, to 5,000 miles and 700. You see immediately which assumption matters most, and those are the ones worth checking.

• Write your assumptions down before you multiply.
• Keep units visible so they cancel properly, such as miles ÷ (miles per gallon) = gallons.
• Expect the answer to be good to within a factor, not a decimal place.

Size the error of an estimate

percent error = (estimate − exact) ÷ exact × 100A positive result means the estimate was too high.

For 48 × 19, the estimate of 1,000 against the exact 912 gives (1,000 − 912) ÷ 912 × 100 = 9.6%. Decide in advance how much error you can tolerate. A tip can be off by a few cents, but a load limit or a figure on a tax form cannot.

The same percent approach gives quick mental answers. To find 15% of a bill, take 10%, add half of it, and you are done; the article on how to work out a tip without a calculator shows the method in detail. When you want the exact figure afterwards, the percentage calculator and the percent error calculator confirm both the answer and how far off your estimate was.

A simple routine

1. Before calculating, say roughly what size the answer should be.
2. Do the exact calculation.
3. Compare the two. If they differ by a factor of 10 or more, look for a decimal-point or unit error.
4. If they differ by 10% or so, check whether your rounding explains it.