Estimating and Sense-Checking

Most mistakes with numbers are not hard maths. They are a slipped decimal point or a mistyped digit. A quick estimate before you calculate lets you spot an answer that cannot be right, and gives you confidence when you have no calculator at all.

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Round first, then calculate

• Mr Mathematics gives a reliable routine: round every number to 1 significant figure first, then calculate with the rounded values.
• To round to 1 significant figure, keep the first non-zero digit, round up if the next digit is 5 or more, and fill the other places with zeros. Mr Mathematics' examples: 578 becomes 600, and 0.0472 becomes 0.05.
• Oak National Academy's example: 6,523 rounds to 7,000. This is quicker but less precise than rounding to 2 significant figures.
• Mr Mathematics estimates 71 x 32.4 / 4.92 squared as 70 x 30 / 5 squared = 84. The exact answer is about 95, so the estimate shows the right size of answer.
• Use the 'approximately equal' sign (Oak) to remind yourself an estimate is not exact.

Everyday estimating

• Shopping basket: items at AED 12.75, 8.50, 23.95, 4.25, 17.80 and 6.99. Round each to the nearest dirham (13, 9, 24, 4, 18, 7) and add: about AED 75. The exact total is AED 74.24.
• If the till says AED 742.40, your estimate tells you instantly that something is wrong.
• Tip check: 15 percent of AED 240 is AED 36. An answer of AED 360 means the decimal point has slipped.
• Screen time: 2.5 hours a day for a year is 2.5 x 365 = 912.5 hours, about 38 whole days. Estimates like this can be surprising and useful.

Back-of-the-envelope (Fermi) estimates

• Wikipedia describes Fermi problems, named after the physicist Enrico Fermi, as estimation problems solved with justified guesses and rough calculations.
• Break a big question into smaller ones you can guess. 'How much water does my family use for drinking in a year?' becomes people x litres per person per day x 365.
• Wikipedia explains that over-estimates and under-estimates in different steps tend to cancel out, so the final estimate is often closer than you would expect.
• Aim to get the right scale (tens, hundreds, thousands), not an exact figure.

Spotting an answer that cannot be right

• Mr Mathematics notes that an estimate is a quick check that a calculator answer is sensible; if they are very different, you may have mistyped a value or misplaced a decimal point.
• Ask: is it the right size? A taxi fare of AED 3,500 or a fuel bill of AED 0.48 is clearly wrong.
• Ask: is it the right direction? A discount should make a price smaller; adding VAT should make it bigger.
• Ask: are the units sensible? Speeds in km/h, fuel in litres, time in hours and minutes.
• Oak National Academy suggests using known benchmarks: a hand is about 10 cm wide and a guitar is about 1 m long. Use similar mental benchmarks for prices you know well.

Building number confidence

• The Department for Education's Education Hub says ability is not fixed and everyone can improve; praising effort and working out steps helps.
• Estimate first, every time. It turns maths into a quick habit rather than a test.
• Say your reasoning out loud with a friend or family member; explaining a method strengthens it.
• Be kind to yourself after mistakes: the point of sense-checking is to catch them, and everyone makes them.