Find ten percent first

Ten percent is the number with its decimal point moved one place to the left, and everything else is built from that. Many people find ten percent first and then scale it: double it for twenty percent, halve it for five, halve it and add it back for fifteen, add the five to the twenty for twenty-five. Others say the same, and some say the tricks get their most practical use on a restaurant bill. Some people build from other fractions when they are closer: a third for thirty-three percent, a half for fifty, a quarter for twenty-five, and then add or subtract a ten. Some people round first — the bill to a convenient number — because some people warn this is estimation anyway and a whole number is faster to slide. Some people 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, and the harder half of a hard sum can often be made the easy half. The mistakes people caught here are the reason this page is careful with the decimal: the person who introduced the trick worked ten percent of 72.64 as 7.20, and was corrected — it is 7.26; and another person said to ‘remove a zero’ to divide by ten, and was corrected — removing a digit from 1081 gives 181, which is wrong, while moving the point gives 108.1. Slide the point; do not delete anything. Some people find the method unintuitive despite its simplicity, which is fair: the first few times, write the ten percent down.

Fives, elevens and squares, and people give them as theirs. Some people say dividing by five is doubling and then sliding the point one place left — 340 divided by 5 is 680 slid to 68 — and some people say multiplying by five is the mirror: halve, then slide the point right, so 84 times 5 is 42 with a zero, 420. One person points out, for the odd numbers: half of an odd number ends in .5, and you carry that as a five, not as a zero — 37 times 5 is 18.5 slid to 185; and one person says to think of it as moving the point rather than ‘adding a zero’, because with a decimal in play adding a zero is wrong. Some discussions break awkward multiplications into pieces: times eleven is times ten plus the number itself — 23 times 11 is 230 plus 23, 253; times twelve is times ten plus double; and anything can be split by its factors. Some people give the square of a number ending in five: take the digits before the five, multiply by the next number up, and write 25 after it — 35 squared is 3 times 4, then 25: 1225; 65 squared is 6 times 7, then 25: 4225. One person keeps an anchor table — the fifteens — and finds neighbours by adding or subtracting once: 14 times 6 is 15 times 6 minus 6. Some people say what these are, which helps rather than hurts: the distributive law and a couple of algebraic identities with numbers in place of letters; there is no magic, only a route through easy steps. This page checked every worked example on it.

When exactness matters, when the tax rate is not what the trick assumes, and when the numbers get big. Some people warn the method is estimation: rounding the bill first means the answer is close, and for a tip or a sale that is the point, but a deposit, a tax return or a contract wants the calculator, and some people say so without shame. One person warns 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. Some people warn, for the US reader, that the tips printed on a receipt 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; one person warns tipping on the after-tax total simply tips more — check which line the slip used, and decide on purpose. Some people warn the tricks fall apart on three- and four-digit numbers unless they sit near a round hundred or thousand, and one describes the loop where you cannot finish the sub-sum the trick created without another trick. One person warns: schools may still want the standard method shown, so a child who uses the shortcut should write the working too. One person gives the mental-maths error that actually changes decisions, and this page carries it as one clause because it belongs on any page about percentages: 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. Two small ones from individual people: if the total in your account is out by an amount divisible by nine, you have swapped two digits somewhere; and to add two fractions with one on top, add the bottoms for the new top and multiply them for the new bottom.

Yes, and people argue about what that means, so here are the sides and the decider. In the main disagreement, some people say specific tricks are useful shortcuts for anyone who freezes at arithmetic or wants a quick answer at a table; many people say the tricks are ordinary arithmetic dressed up, that a person who knows the multiplication table and the standard method never needs them, and that learning the fundamentals beats collecting shortcuts. Other discussions say the same from both sides — in one, some say it is a genuine help to people who are not good with numbers, some that it is primary-school arithmetic; in another, people argue whether it is redundant algebra or a shortcut people do not find on their own; a third argues whether mental arithmetic matters at all now that everyone carries a calculator. Some people say the difficulty is only lack of practice and that doing small sums daily makes them natural; others say to open the calculator app and think no more about it. People’s own decider is honest and this page adopts it: who you are. If you were drilled in tables and can see 15 percent of 84 without effort, this page is not for you, and the people who call it ordinary arithmetic are right that you never needed it. If you are the person who has paid the printed tip for years to avoid the arithmetic, or pretended to check your phone at the rail, the people who call the tricks a shortcut are right that a route through ten percent is a real help, and it is also — this page’s own reading — a way into the practice the other side recommends. One person who teaches says the tricks are better taught as ways of thinking than as rules, and that they help children understand why the standard method works; another says they should be taught earlier than they are. People would not think less of you for opening the app.

They exist, and some people say they are ordinary long multiplication in a different costume. A family of pattern tricks — for multiplying two numbers just under a hundred, or for crossing digits in a lattice — is argued in many discussions here, and what people pointed out settles what can be settled: the ‘vertical and crosswise’ method is the same four small multiplications as the standard method, added in a different order, and the lattice method is standard multiplication drawn differently and summed on the diagonals. Some people warn the tricks work only for numbers near a round hundred and get harder than the ordinary way for others, and one person works through a pair — 66 times 58 — where the trick needs an implicit carry the standard method does not. One person says they are useful for competitions and rarely for life; in another discussion, some people say they are easier and should be taught in schools, and some people say they are less general than what schools teach. This page’s own line: if one of them clicks for you, use it, and check the answer the ordinary way until it stops surprising you; the ten percent move from the first question is the one that will actually be needed this week. The miles-to-kilometres trick one person passes around — double four times, then divide by ten — is not this page.