How do I measure, convert and work out percentages in my head?

Four pages in this library are about the sum you need while somebody is talking to you. Find ten percent first is the receipt on the table and the sale sign on the rail. Measure it with your hand is the shop with no tape measure. Double it and add thirty is the forecast in a country where the numbers are the other way round. One mark for feet, two for inches is the pen in the air over 5′10. The first three say plainly that they are estimation and name the one place where estimation is not good enough; the fourth says nothing on it can hurt you.

Find ten percent first, then scale it. Find ten percent first says eight separate conversations, the widest agreement in its set, use one move and nothing more: slide the decimal point one place to the left, so ten percent of 72.60 is 7.26, then double it for twenty, halve it for five, halve it and add it back for fifteen, add the five to the twenty for twenty-five. Two conversations round the bill first, because a whole number slides faster and the answer is close enough for a tip or a sale. Two more give the swap that surprises people: x percent of y is the same as y percent of x, so eight percent of fifty becomes fifty percent of eight, which is four. The page is careful with the decimal because of two corrections in its conversations: the account who introduced the trick worked ten percent of 72.64 as 7.20 and was corrected to 7.26, and another said to ‘remove a zero’ to divide by ten, and was corrected, because removing a digit from 1081 gives 181 while moving the point gives 108.1. Slide the point; do not delete anything. Four separate conversations add the fives: to divide by five, double and slide the point left, so 340 divided by 5 is 680 slid to 68; two say to multiply by five, halve and slide right, so 84 times 5 is 420, and one correction says half of an odd number ends in .5 and you carry that as a five. Two conversations give the square of a number ending in five: the digits before the five multiplied by the next number up, then 25 after it, so 35 squared is 3 times 4, then 25, which is 1225. The page says it checked every worked example on it. Whether any of this is worth knowing is argued three accounts to seven, the seven saying it is ordinary arithmetic dressed up and the tables beat it, and the page adopts the accounts’ decider, which is who you are: the person who was drilled did not need the page, and the person who has paid the printed tip for years to avoid the sum does. Two conversations say just open the calculator, and the page says the people there would not think less of you for it.

Where the percentage trick goes wrong. Two cautions on that page say the method is estimation: rounding first means the answer is close, not exact, so when the figure matters to the penny, a contract, a deposit, a tax return, the page’s own line is to use a calculator without embarrassment. Two cautions, for the US reader with a receipt in hand, say the tips printed on a slip as suggestions are usually worked on the total after tax while the percentage you intend is of the total before it, so the printed figures run higher than yours; a third caution says tipping on the after-tax total simply tips more, so check which line the slip used, and decide on purpose. One caution says a popular shortcut, triple the sales tax on a US receipt to get the tip, works only where the tax rate happens to be a third of the tip you meant, and is wrong everywhere else. Two cautions say the tricks fall apart on three- and four-digit numbers unless they sit near a round hundred or thousand. And one account gives what the page calls the mental-maths error that actually changes decisions: a headline that says something ‘increases the risk by eighty percent’ is describing a relative increase, the new chance depends entirely on how small the old one was, and eighty percent more of a tiny number is still a tiny number.

Measure your own hand once, and carry the two numbers as your tape. Measure it with your hand says six separate conversations use the stretched hand, thumb-tip to little-finger-tip, as their long unit, roughly eight or nine inches on the adults who measured, and six use the thumb’s width or first joint as their inch; two add elbow to fingertip as about a foot, the old cubit. The page’s exercise is to measure those two things tonight with a ruler and write both on your phone, and its caution is that it is your hand that is calibrated, not hands in general. The accounts are plain that all of it is rough: one caution two accounts strong says body parts vary and nothing about them is standardised, single cautions add that hands change with weight, that a finger with an old injury will not measure true, and that your dominant hand and your other hand are not quite the same size, and two accounts say a measurement memorised young leads to persistent errors, because you grew and the number did not. Three separate conversations reach for the banknote in the wallet instead, because it does not grow; several call the US dollar bill six inches and one correction fixes it, which the page checked against the Bureau of Engraving and Printing: 6.14 inches long and 2.61 wide, the same for every denomination. Two cautions, two accounts each, say older notes can differ in size through wear or changes in manufacture, and that sizes differ between countries and eras. For a room, two conversations say learn your stride and count paces, with one caution that stride varies a good deal from day to day; one account takes the distance from the far shoulder to the fingertips of an outstretched arm as roughly a metre. If you need a precise figure, the accounts say a coin or a ruler, not a finger, and the page agrees.

The fingers and the sun, with the line that comes first. The hand page carries one safety line, for the trick that counts the fingers between the bottom of the sun and the horizon to guess how long until sunset: do not look at the sun. Five accounts across two cautions say it, one rated high; hold your hand up so it covers the sun, look at the hand, and never stare past it. Three separate conversations describe the trick, roughly fifteen minutes of daylight for each finger stacked sideways along an outstretched arm, and the corrections and limits are many: measure along the sun’s actual path down, not straight down; one contraindication and three cautions say the method fails in high latitudes, where the sun sets at a shallow angle or in summer does not set at all; one caution says you need a visible horizon, which mountains take away; and one account settles it for hikers as a quick, low-tech estimate for planning camp or the hours of walking left, not a precise time.

Double the Celsius and add thirty. Double it and add thirty says that is it: twenty becomes seventy against a true sixty-eight, ten becomes fifty, which is exact, thirty becomes ninety against a true eighty-six, and across the range of temperatures a day actually is you are within a few degrees. Going the other way, subtract thirty, then halve. The rule drifts upward as the number climbs, and the page puts the one place that matters above everything else: do not use this rule for anybody’s body temperature. It is built to be accurate around weather numbers, and a fever of 38.5°C comes out of it as 107°F when the real answer is 101.3°F, which the page calls the gap between a call to the doctor tomorrow and an ambulance tonight, in the frightening direction. For a temperature that matters, do the proper sum, multiply by 9, divide by 5, add 32, or use a thermometer that shows the unit you think in. Some people skip the arithmetic and memorise landmarks: sixteen Celsius is sixty-one Fahrenheit and twenty-eight is eighty-two, with the digits swapped, which the page calls a coincidence and an extremely useful one, and there is a rhyme the page passes on, thirty is hot, twenty is nice, ten is cold, zero is ice. For miles and kilometres, the page carries the famous Fibonacci trick, five miles is eight kilometres, eight is thirteen, thirteen is twenty-one, checked as genuinely accurate, alongside the practical argument against it from several people at once, that it only lands on the numbers in the sequence, is poor at the small end, and needs you to break any other distance into Fibonacci pieces and add them up; the boring method is to multiply by 1.6, or add half and then a tenth of the original, so ten miles is sixteen kilometres against a true 16.09. For speed limits, halve the kilometres and add the first digit, so 50 km/h becomes 30 mph against a true 31.07, and many cars have both scales printed on the speedometer. The page says somebody in its conversations pointed out that none of this is mathematics, it is rearranged arithmetic, and that they were right; the value is that these versions survive being done in your head, in a car, in a queue, while somebody is asking you a question, which the correct formula does not.

One mark for feet, two for inches. One mark for feet, two for inches says four conversations remember it by syllables, feet, one, inch-es, two, and a correction backed by four accounts says that is a coincidence that happens to agree: the real reason is order, the first unit named gets one mark and the second gets two, the same way minutes take one mark and seconds two on a clock or a compass. Two cautions, five accounts between them, say the syllable version fails for people who say ‘inch’ as one beat, or ‘tomato’ differently in the related trick for feet in a mile, because pronunciation is not a stable foundation; hold the order and let the syllables be a crutch. For which months have thirty-one days, six separate conversations count along the knuckles of a fist and five recite the rhyme, thirty days hath September, April, June and November; the exchange runs five accounts to six and the accounts’ decider is how your memory works, by hand or by ear. Two conversations add the international line the page follows: write the month as a word, 14 Oct 2014, so that a date read abroad is not the fourteenth month. Three conversations give the one trick on that page that drew no argument: add the digits, and the number is divisible by three if the sum is. Five conversations count to thirty-one on one hand in binary, thumb one, index two, middle four, ring eight, little finger sixteen, and four accounts and two more say some of the hand shapes look like rude gestures, so count under the table or on a flat hand. And the page carries its own warning about tricks: two cautions say a mnemonic can be counterproductive when it is harder to remember than the fact it stands for, and the page says to drop it if it is.

Who these pages are not for. The percentage page is not for the figure that matters to the penny, and its tax cautions are US. The hand page says nothing on it is a substitute for a real measurement when a real measurement matters, and its daylight trick fails at high latitudes, in midnight sun, without a horizon, and if you look at the sun. The temperature page is not for anybody’s body temperature, for cooking temperatures, medicine or engineering, or anything where being four degrees out is a problem, and its own honest note is that it is a set of tools for being roughly right quickly, which is nearly always what is wanted and once in a while is not. And the feet-and-inches page says that if you live in a country that measures in centimetres, its first question is not yours, and that which system you measure in is decided by where you live and not by a tip.