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Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything
Rob Eastaway
‘Another terrific book by Rob Eastaway’ SIMON SINGH ‘A delightfully accessible guide to how to play with numbers’ HANNAH FRY How many cats are there in the world? What's the chance of winning the lottery twice? And just how long does it take to count to a million? Learn how to tackle tricky maths problems with nothing but the back of an envelope, a pencil and some good old-fashioned brain power. Join Rob Eastaway as he takes an entertaining look at how to figure without a calculator. Packed with amusing anecdotes, quizzes, and handy calculation tips for every situation, Maths on the Back of an Envelope is an invaluable introduction to the art of estimation, and a welcome reminder that sometimes our own brain is the best tool we have to deal with numbers.





Copyright
HarperCollinsPublishers 1 London Bridge Street London SE1 9GF
www.harpercollins.co.uk (http://www.harpercollins.co.uk)
First published by HarperCollinsPublishers 2019
FIRST EDITION
© Rob Eastaway 2019
Cover design by Andrew Davis © HarperCollinsPublishers Ltd 2019 Cover photograph © Shutterstock.com (http://Shutterstock.com)
A catalogue record of this book is available from the British Library
Rob Eastaway asserts the moral right to be identified as the author of this work
All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, non-transferable right to access and read the text of this e-book on screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
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Source ISBN: 9780008324582
Ebook Edition © September 2019 ISBN: 9780008324599
Version 2019-08-30

CONTENTS

1  Cover (#ube9b5bb8-b01d-57f4-8048-036dba1a0d71)
2  Title Page (#u893be14e-95f5-5740-afd9-b8d58d8996fc)
3  Copyright (#u576a6c51-6ba3-5f83-b15b-ad9155b62a18)
4  Prologue: How Many Cats? (#udba157d3-b9d4-53df-a26f-8169e146f6c4)
5  THE PERILS OF PRECISION (#ue3c4e76b-a510-5b47-8813-fd8411bc5b08)
6  TOOLS OF THE TRADE (#u3cfd0fa3-b485-5582-8ac7-19d2231b1390)
7  EVERYDAY ESTIMATION (#litres_trial_promo)
8  FIGURING WITH FERMI (#litres_trial_promo)
9  LAST WORD: WHEN THE ROBOTS TAKE OVER (#litres_trial_promo)
10  Appendix (#litres_trial_promo)
11  Answers and Tips (#litres_trial_promo)
12  Endnotes (#litres_trial_promo)
13  Acknowledgements (#litres_trial_promo)
14  About the Author (#litres_trial_promo)
15  About the Publisher (#litres_trial_promo)

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1  Cover (#ube9b5bb8-b01d-57f4-8048-036dba1a0d71)
2  Frontmatter (#u893be14e-95f5-5740-afd9-b8d58d8996fc)
3  Begin Reading (#u893be14e-95f5-5740-afd9-b8d58d8996fc)
4  Table of Contents (#u4ad50bbf-de7a-518c-8447-bda80dc9bac9)

PROLOGUE
HOW MANY CATS?
A few years ago, at a school event, I asked the audience of teenagers to submit some estimation questions that I would then attempt to answer live on stage. One pupil posed this simple question: ‘How many cats are there in the world?’
Cats are always a popular topic, so I took it on.
My thinking went like this:
Let’s assume that most cats are domestic.
Some people have more than one cat, but usually a household has only one cat, if any at all.
In the UK, and thinking of my own street as an example, it seems reasonable to suppose that there might be one cat in every five households.
And, if a household contains on average two people, that means there is one cat for every 10 people.
So, with 70 million people in the UK, let’s say that there are, perhaps, seven million cats in the UK.
So far, so good. But what about the number of cats in the rest of the world? It seems unlikely that cats are as popular in countries like India or China as they are in the UK (although what would I know? Remember, this is purely guesswork on my part), therefore, I’d expect the ratio across the world to be smaller than it is in the UK – maybe one cat for every 20 people?
So, with eight billion people in the world, that suggests there are maybe:
8 billion ÷ 20 = 400 million cats.
It doesn’t seem an outrageous number.
That was the figure I suggested, anyway.
A member of the audience put his hand up.
‘The real number is 600 million,’ he said.
‘Really – how do you know?’
‘I just looked it up online.’
So that’s it: no need to come up with an answer; it’s already been done.
If it’s really that simple, we can forget about doing estimations altogether. With just a few clicks on Google you will probably find a statistic to answer every question you could possibly think of.
Except for one crucial thing.
Where did the person who published that figure of 600 million online get their number from? No one, I’m quite sure, has gone around the world doing a census of cats. The figure of 600 million is an estimate. It might be an estimate that is based on a slightly more scientific method than mine, using rigorous surveys and cross-checks, but it’s just as likely that the figure cited online came from somebody who did a back-of-envelope calculation like the one I just described. Or, perhaps they simply invented a figure that suited their agenda. There’s no reason to believe that their number is more reliable than mine – indeed, it might be less reliable.
Once a statistic like this is out there, published in a newspaper, quoted on a website, it becomes ‘fact’, and it can be re-quoted so often that the source may go unquestioned and very quickly be forgotten altogether.
It’s an important reminder that the majority of statistics published anywhere are estimates, many of them worked out on the equivalent of the back of an envelope. When back-of-envelope calculations produce a very different answer from the one that’s been put forward, it doesn’t necessarily mean that the estimate is wrong. It means rather that the published figure deserves more scrutiny.
We tend to think of maths as being an ‘exact’ discipline, where answers are right or wrong. And it’s true that there is a huge part of maths that is about exactness.
But in everyday life, numerical answers are sometimes just the start of the debate. If we are trained to believe that every numerical question has a definitive, ‘right’ answer then we miss the fact that numbers in the real world are a lot fuzzier than pure maths might suggest.
I’ve realised in writing this book that there is a kind of paradox. On the one hand, I want to argue that approximate numbers can often be more informative, and more trustworthy, than precise numbers. Yet at the same time, in order to be able to produce those approximate answers, it’s essential to know how to do some calculations exactly. Your basic times tables, for example. Concrete, exact maths is the foundation for the woolly numbers that we have to deal with in everyday life.
I’ve divided the book into four sections.
In the first section, I explore how precise numbers can be misleading, and why it’s good not to be entirely dependent on a calculator.
The second section includes the arithmetical techniques and the other knowledge that is an essential foundation if you want to embark on back-of-envelope calculations. This includes a refresher on how to do arithmetic that you may not have needed to practise since you left primary school, as well as short cuts that you probably never encountered there.
The rest of the book shows how to use these techniques to tackle problems, from everyday conversions, to more serious issues like helping the environment. And at the end, there is a collection of so-called Fermi questions: quirky and esoteric challenges to come up with a reasonable answer based on very little hard data.
Back-of-envelope maths is an important and valuable life skill. But that’s not its only benefit. Many of us also indulge in it simply because it’s a fun and stimulating exercise that keeps the brain sharp.

1
THE PERILS OF PRECISION
ENVELOPES VERSUS CALCULATORS
I don’t know when the backs of envelopes first became a popular place for jotting rough-and-ready calculations. Was it before or after people started to use ‘the back of a fag packet’, or in the USA, the back of a napkin?
Regardless of where the expression was first coined, the back of an envelope1 (#litres_trial_promo) has come to symbolise any sort of rough-and-ready type of calculation that gives an indication of what the right answer will be.
It is the tool that people in business use for quickly checking the viability of a new project. Engineers use it to check if a proposed solution is likely to work. And commentators on statistics use it to help make sense of the myriad numbers that are thrown out by politicians, ‘expert’ pundits and marketers.
On a more mundane level, it’s the maths you might use every day to ensure you aren’t getting ripped off by a so-called ‘deal’ that turns out to be anything but.
It is also maths and arithmetic that can be done without needing to resort to a calculator.
But wait a minute. Maths without a calculator? To many people, this notion seems quaintly old-fashioned, or even masochistic. Why grapple with manual or mental calculations when most of us have a phone (with a calculator) readily to hand almost all of the time?
This is not an anti-calculator book. Calculators are indispensable tools that have enabled us to do in seconds what used to take minutes, hours or even days. If you need to know exactly what £31.40 × 96 is, then unless you are a savant or somebody with plenty of time on your hands, a calculator is the only sensible option for working it out. And I’m probably typical in usually having a calculator – or a spreadsheet – to hand if I’m doing my tax return, or totting up expenses after a work-related trip.
But much of the time we don’t need to know the exact answer. It’s an approximate figure that matters. The point of back-of-envelope maths is to help see the bigger picture behind numbers.
Suppose a sales team has a target of £10,000. If they report that they have sold 96 units at £31.40 each – that’s roughly:
100 × £30 = £3,000 revenue.
That’s massively short of the £10,000 target, even if the estimate is out by a few per cent.
When the government announces a £1 billion increase in health spending, is that significant? Spread between 50 million people? It won’t be exactly one billion pounds of course, nor will it be spread evenly between 50 million people, but with back-of-envelope maths, we can work out it will represent an average of something nearer to £20 (i.e. hardly anything) than £200 per person.
Of course, even these simplified calculations can be done on a calculator. But the reality is that they rarely are.
The argument: ‘Who needs to do arithmetic when we all have calculators?’ is usually a red herring. In situations where a calculation is not essential, most of us do it in our heads or on the back of an envelope, or don’t do it at all.
And there are some who use their ability to figure things mentally to their advantage. I have a friend who made his fortune as a wheeler-dealer in finance. I asked him to share some advice.
‘I have two tips for succeeding when negotiating a deal with somebody,’ he said. ‘The first is: learn how to be able to read upside down, so that you can decipher the documents of the person opposite you. And my second is: be able to do the calculations faster than they can.’

TEST YOURSELF
How is your arithmetic without the aid of a calculator? Try these 10 questions. There is no time pressure, and you’re allowed to use pencil and paper if you want. As you do these questions, you might want to think about how you do them. Are you recalling facts you’ve memorised? Do you use a pencil-and-paper method?
(a) 17 + 8
(b) 62 – 13
(c) 2,020 – 1,998
(d) 9 × 4
(e) 8 × 7
(f) 40 × 30
(g) 3.2 × 5
(h) One-quarter of 120
(i) What is 75% as a fraction?
(j) What is 10% of 94?
Solutions
I can still remember the thrill when I first got a calculator of my own. It was made by Commodore, and had red LED digits and buttons that made a satisfying click when you pressed them. It was a Christmas present, and I was 16 years old. I was captivated. Just being able to enter a number like 123456 and press the square root button was enough to send a tingle of excitement down my spine, as I gazed at all those digits after the decimal point. I’d never seen numbers to such precision before.
There were two things that came out of the arrival of cheap calculators.
The first was that we could all now do calculations that we would never have conceived of doing before. It was empowering, liberating and gave us a chance to see the bigger picture of mathematics without getting bogged down in the nitty-gritty of calculation.2 (#litres_trial_promo)
The second thing that happened was that we could now quote answers to several decimal places. The square root of 83? Certainly, sir, just give me one second – and how many digits would you like after the decimal point?
What could possibly go wrong?

SPURIOUS PRECISION
A tourist in a natural history museum was very impressed by the skeleton of a Tyrannosaurus Rex.
‘How old is that fossil?’ she asked one of the guides.
‘It’s 69 million years and 22 days,’ said the guide.
‘That’s incredible, how do you know the age so precisely?’ asked the tourist.
‘Well, it was 69 million years old when I started working at the museum, and that was 22 days ago,’ replied the guide.
The thoughtless precision of the museum guide in this old joke nicely illustrates why there is no point in stating a number to several figures if the overall measurement is only a rough estimate. Yet it is a mistake that is made time and again when presenting and interpreting numbers in everyday life.
Quoting a number to more precision than is justified is often called spurious accuracy, though it should really be called spurious precision and we will encounter it several times in this book. It is one of the strongest arguments against the unthinking overuse of calculators. The fact that you can work out numbers to several decimal places at the touch of a button doesn’t mean that you should.

PRECISION VERSUS ACCURACY
The words precision and accuracy are often used interchangeably, to indicate how ‘right’ a measurement or number is. It is certainly possible for a number to be accurate and precise; for example: 74 × 23.2 = 1,716.8.
But used mathematically, precision and accuracy mean different things.
‘Accuracy’ is an indication of how close you are to the right answer. Suppose we are playing darts. I throw a dart at a dartboard and just miss the bullseye. My throw was quite accurate, but if you then throw and hit the bullseye, your throw was more accurate than mine. Likewise … if I tot up the items in my shopping basket and estimate that the total will be £65 while you reckon it will be £70, and the bill turns out to be £69.43, then you were more accurate than I was.
Precision, on the other hand, is an indication of how confident you are in a number to a particular level of detail, so that you or somebody else would come up with the same figure if you did a measurement or calculation again. If you think the shopping basket will add up to £69, you are confident that you are right to the nearest pound; but if you suggest the bill will be £69.40, you are being more precise, and are confident your figure is right to the nearest 10 pence. Even more precise is £69.41. In maths terms, precision is about how manysignificant figures(this is an important concept – please see here (#litres_trial_promo)) you can quote a number to.
As a society, we put a lot of faith in precision. If we see a number such as 84.36, we tend to believe that the person who produced the number is confident of that figure to the second decimal place. We might even honour them with the label of ‘expert’ because they were able to produce a number to such precision. But people who produce ‘precise’ figures often abuse our trust, and accidentally or deliberately they imply a level of confidence that is not justified. When we read that 59,723 people attended an Arsenal football match, we are being led to believe that an exact tally was taken as fans went through the turnstiles. So when we discover that the actual number in the ground was closer to 50,000, we feel like we have been conned.3 (#litres_trial_promo)
When it comes to the use of numbers to interpret the world, accuracy is more important than precision. After all, a measurement that is accurate but not precise can still be helpful. But a measurement that is precise but inaccurate is not just unhelpful, it can sometimes be dangerous.
One of the unintended consequences of calculators is that they will give answers to as many decimal places as will fit on the screen – and in doing so, they tempt us to work to a level of precision that is often not justified.

DRESSING UP NUMBERS WITH DRESSAGE
In 2012 London hosted the Olympics. People throughout Britain celebrated as gold medallists stepped up on the podium in all sorts of sports in which previous GB athletes had rarely excelled.
There was particular joy when Charlotte Dujardin, who had worked her way up from being a stable girl to becoming an elite equestrian, claimed Britain’s first ever gold medal with her horse Valegro in the dressage event.
Dujardin’s winning score from the judges was an impressive 90.089%.
Sportspeople often talk about ‘giving it that extra ten per cent’, but in this case it seemed as if Dujardin had fine-tuned this, so she could put in that extra 10.001%. What was it that made her performance better than somebody who got, say, 90.088% instead?
To understand where her three-decimal-place score came from, we need to look at how the judges allocated marks in that competition.
In the dressage event at the London Olympics, the competitors were required to take their horse through a series of movements, which would be assessed by seven judges seated around the arena so they could view from different angles. The judges were scoring under 21 headings: 16 of them were ‘technical’ marks given for how well specific movements were carried out, and five of them ‘artistic’, applying to the overall performance, with descriptions such as ‘Rhythm, energy and elasticity’ and ‘Harmony between horse and rider’ (yes, really).
Each item was marked out of 10, with half-marks allowed, but some scores were then given more weighting, and the five artistic scores were all multiplied by four. In total, each judge could give up to:
240 technical marks + 160 artistic merit marks
= 400 marks total.
It meant each rider had a total of 7 × 400 = 2,800 points to play for.
There is an element of subjectivity when assessing how good a horse’s performance has been, so it is hardly surprising that the judges don’t all give the same mark. For a particular technical movement, one judge might give it an ‘8’, while another spots a slightly dropped shoulder and reckons it’s a ‘7’. In fact, in Dujardin’s case, the judges’ total marks ranged from 355 to 370, and when added together she got a total of 2,522.5 points out of a maximum possible 2,800.
And this is where the percentage comes in, because her 2,522.5 total was then divided by 2,800 to give a score out of one hundred, a percentage:
2,522.5 ÷ 2,800 = 90.089%.4 (#litres_trial_promo)
Well, actually that’s not the exact number. It was really 90.089285714285714 … %.
Indeed, this number never stops, the pattern 285714 repeats for ever. This is what happens when you divide a number by a multiple of 7. So Dujardin’s score had to be rounded, and the authorities who were responsible for the scoring system decided to round scores to three decimal places.
What would have happened if Dujardin had been awarded half a mark less by the judges? She would have scored:
2,522 ÷ 2,800 = 90.071%.
In other words, the precision of her actual score of 90.089 was misleading. It wasn’t possible to score any other mark between 90.089% and 90.071%, Dujardin didn’t give it an extra 0.001%, but rather she gave it that extra 0.018%. Quoting her score to two decimal places (i.e. 90.09%) was enough.
The second decimal place is needed to guarantee that two contestants with different marks don’t end up with the same percentage, but it still gives a misleading sense of the accuracy of the scoring. In reality, each judge ‘measures’ the same performance differently. A half-mark disagreement in the artistic score (which is then multiplied by 4, remember) shifts the overall mark by 0.072%. And the actual discrepancies between the judges were bigger than that. For ‘Harmony between horse and rider’ one judge marked her 8 out of 10 while another gave her 9.5 out of 10.

A NUMBER IS ONLY AS STRONG AS ITS WEAKEST LINK
There’s a time and a place for quoting numbers to several decimal places, but dressage, and other sports in which the marking is subjective, isn’t one of them.
By using this scoring system, the judges were leaving us to assume that we were witnessing scoring of a precision equivalent to measuring a bookshelf to the nearest millimetre. Yet the tool they were using to measure that metaphorical bookshelf was a ruler that measured in 10-centimetre intervals. And it was worse than that, because it’s almost as if the judges each had different rulers and, on another day, that very same performance might have scored anywhere between, say, 89% and 92%. It was a score with potential for a lot of variability – more of which in the next section.
All of this reveals an important principle when looking at statistical measurements of any type. In the same way that a chain is only as strong as its weakest link, a statistic is only as reliable as its most unreliable component. That dinosaur skeleton’s age of 69 million years and 22 days was made up of two components: one was accurate to the nearest million years, the other to the nearest day. Needless to say, the 22 days are irrelevant.

BODY TEMPERATURE A BIT LOW? BLAME IT ON SPURIOUS PRECISION
In 1871, a German physician by the name of Carl Reinhold Wunderlich published a ground-breaking report on his research into human body temperature. The main finding that he wanted to publicise was that the average person’s body temperature is 98.6 degrees Fahrenheit, though this figure will vary quite a bit from person to person.
The figure of 98.6 °F has become gospel,5 (#litres_trial_promo)the benchmark body temperature that parents have used ever since when checking if an unwell child has some sort of fever.
Except it turns out that Wunderlich didn’t publish the figure 98.6 °F. He was working in Celsius, and the figure he published was 37 °C, a rounded number, which he qualified by saying that it can vary by up to half a degree, depending on the individual and on where the temperature is taken (armpit or, ahem, orifice).
The figure 98.6 came from the translation of Wunderlich’s report into English. At the time, Fahrenheit was the commonly used scale in Britain. To convert 37 °C to Fahrenheit, you multiply by 9, divide by 5 and add 32; i.e. 37 °C converts to 98.6 °F. So the English translation – which reached a far bigger audience than the German original, gave the figure 98.6 °F as the human norm. Technically, they were right to do this, but the decimal place created a misleading impression. If Wunderlich had quoted the temperature as 37.0 °C, it would have been reasonable to quote this as 98.6 °F, but Wunderlich deliberately didn’t quote his rough figure to the decimal place. For a figure that can vary by nearly a whole degree between healthy individuals, 98.6 °F was (and is) spurious precision. And in any case, a study in 2015 using modern, more accurate thermometers, found that we’ve been getting it wrong all these years, and that the average human temperature is 98.2 °F, not 98.6 °F.

VARIABILITY
In the General Election of May 2017, there was a shock result in London’s Kensington constituency. The sitting MP was a Conservative with a healthy majority, but in the small hours of the Friday, news came through that the result was too close to call, and there was going to be a recount. Hours later, it was announced that there needed to be a second recount. And then, when even that failed to resolve the result, the staff were given a few hours to get some sleep, and then returned for a third recount the following day.
Finally, the returning officer was able to confirm the result: Labour’s Emma Dent Coad had defeated Victoria Borwick of the Conservatives.
The margin, however, was tiny. Coad won by just 20 votes, with 16,333 to Borwick’s 16,313.
You might expect that if there is one number of which we can be certain, down to the very last digit, it is the number we get when we have counted something.
Yet the truth is that even something as basic as counting the number of votes is prone to error. The person doing the counting might inadvertently pick up two voting slips that are stuck together. Or when they are getting tired, they might make a slip and count 28, 29, 40, 41 … Or they might reject a voting slip that another counter would have accepted, because they reckon that marks have been made against more than one candidate.
As a rule of thumb, some election officials reckon that manual counts can only be relied on within a margin of about 1 in 5,000 (or 0.02%), so with a vote like the one in Kensington, the result of one count might vary by as many as 10 votes when you do a recount.6 (#litres_trial_promo)
And while each recount will typically produce a slightly different result, there is no guarantee which of these counts is actually the correct figure – if there is a correct figure at all. (In the famously tight US Election of 2000, the result in Florida came down to a ruling on whether voting cards that hadn’t been fully punched through, and had a hanging ‘chad’, counted as legitimate votes or not.)
Re-counting typically stops when it is becoming clear that the error in the count isn’t big enough to affect the result, so the tighter the result, the more recounts there will be. There have twice been UK General Election votes that have had seven recounts, both of them in the 1960s, when the final result was a majority below 10.
All this shows that when it is announced that a candidate such as Coad has received 16,333 votes, it should really be expressed as something vaguer: ‘Almost certainly in the range 16,328 to 16,338’ (or in shorthand, 16,333 ± 5).
If we can’t even trust something as easy to nail down as the number of votes made on physical slips of paper, what hope is there for accurately counting other things that are more fluid?
In 2018, the two Carolina states in the USA were hit by Hurricane Florence, a massive storm that deposited as much as 50 inches of rain in some places. Among the chaos, a vast number of homes lost power for several days. On 18 September, CNN gave this update:
511,000—this was the number of customers without power Monday morning—according to the US Energy Information Administration. Of those, 486,000 were in North Carolina, 15,000 in South Carolina and 15,000 in Virginia. By late Monday, however, the number [of customers without power] in North Carolina had dropped to 342,884.
For most of that short report, numbers were being quoted in thousands. But suddenly, at the end, we were told that the number without power had dropped to 342,884. Even if that number were true, it could only have been true for a period of a few seconds when the figures were collated, because the number of customers without power was changing constantly.
And even the 486,000 figure that was quoted for North Carolina on the Monday morning was a little suspicious – here we had a number being quoted to three significant figures, while the two other states were being quoted as 15,000 – both of which looked suspiciously like they’d been rounded to the nearest 5,000. This is confirmed if you add up the numbers: 15,000 + 15,000 + 486,000 = 516,000, which is 5,000 higher than the total of 511,000 quoted at the start of the story.
So when quoting these figures, there is a choice. They should either be given as a range (‘somewhere between 300,000 and 350,000’) or they should be brutally rounded to just a single significant figure and the qualifying word ‘roughly’ (so, ‘roughly 500,000’). This makes it clear that these are not definitive numbers that could be reproduced if there was a recount.
And, indeed, there are times when even saying ‘roughly’ isn’t enough.
Every month, the Office for National Statistics publishes the latest UK unemployment figures. Of course this is always newsworthy – a move up or down in unemployment is a good indicator of how the economy is doing, and everyone can relate to it. In September 2018, the Office announced that UK unemployment had fallen by 55,000 from the previous month to 1,360,000.
The problem, however, is that the figures published aren’t very reliable – and the ONS know this. When they announced those unemployment figures in 2018, they also added the detail that they had 95% confidence that this figure was correct to within 69,000. In other words, unemployment had fallen by 55,000 plus or minus 69,000. This means unemployment might actually have gone down by as many as 124,000, or it might have gone up by as many as 14,000. And, of course, if the latter turned out to be the correct figure, it would have been a completely different news story.
When the margin of error is larger than the figure you are quoting, there’s barely any justification in quoting the statistic at all, let alone to more than one significant figure. The best they can say is: ‘Unemployment probably fell slightly last month, perhaps by about 50,000.’
It’s another example of how a rounded, less precise figure often gives a fairer impression of the true situation than a precise figure would.

SENSITIVITY
We’ve already seen that the statistics should really carry an indication of how much of a margin of error we should attach to them.
An understanding of the margins of error is even more important when it comes to making predictions and forecasts.
Many of the numbers quoted in the news are predictions: house prices next year, tomorrow’s rainfall, the Chancellor’s forecast of economic growth, the number of people who will be travelling by train … all of these are numbers that have come from somebody feeding numbers into a spreadsheet (or something more advanced) to represent this mathematically, in what is usually known as a mathematical model of the future.
In any model like this, there will be ‘inputs’ (such as prices, number of customers) and ‘outputs’ that are the things you want to predict (profits, for example).
But sometimes a small change in one input variable can have a surprisingly large effect on the number that comes out at the far end.
The link between the price of something and the profit it makes is a good example of this.
Imagine that last year you ran a face-painting stall for three hours at a fair. You paid £50 for the hire of the stand, but the cost of materials was almost zero. You charged £5 to paint a face, and you can paint a face in 15 minutes, so you did 12 faces in your three hours, and made:
£60 income – £50 costs = £10 profit.
There was a long queue last year and you were unable to meet the demand, so this year you increase your charge from £5 to £6. That’s an increase of 20%. Your revenue this year is £6 × 12 = £72, and your profit climbs to:
£72 income – £50 costs = £22 profit.
So, a 20% increase in price means that your profit has more than doubled. In other words, your profit is extremely sensitive to the price. Small percentage increases in the price lead to much larger percentage increases in the profit.
It’s a simplistic example, but it shows that increasing one thing by 10% doesn’t mean that everything else increases by 10% as a result.7 (#litres_trial_promo)

EXPONENTIAL GROWTH
There are some situations when a small change in the value assigned to one of the ‘inputs’ has an effect that grows dramatically as time elapses.
Take chickenpox, for example. It’s an unpleasant disease but rarely a dangerous one so long as you get it when you are young. Most children catch chickenpox at some point unless they have been vaccinated against it, because it is highly infectious. A child infected with chickenpox might typically pass it on to 10 other children during the contagious phase, and those newly infected children might themselves infect 10 more children, meaning there are now 100 cases. If those hundred infected children pass it on to 10 children each, within weeks the original child has infected 1,000 others.
In their early stages, infections spread ‘exponentially’. There is some sophisticated maths that is used to model this, but to illustrate the point let’s pretend that in its early stages, chickenpox just spreads in discrete batches of 10 infections passed on at the end of each week. In other words:
N = 10
,
where N is the number of people infected and T is the number of infection periods (weeks) so far.
After one week: N = 10
= 10.
After two weeks: N = 10
= 100.
After three weeks: N = 10
= 1,000,
and so on.
What if we increase the rate of infection by 20% to N = 12, so that now each child infects 12 others instead of 10? (Such an increase might happen if children are in bigger classes in school or have more playdates, for example.)
After one week, the number of children infected is 12 rather than 10, just a 20% increase. However, after three weeks, N = 12
= 1,728, which is heading towards double what it was for N = 10 at this stage. And this margin continues to grow as time goes on.

CLIMATE CHANGE AND COMPLEXITY
Sometimes the relationship between the numbers you feed into a model and the forecasts that come out are not so direct. There are many situations where the factors involved are inter-connected and extremely complex.
Climate change is perhaps the most important of these. Across the world, there are scientists attempting to model the impact that rising temperatures will have on sea levels, climate, harvests and animal populations. There is an overwhelming consensus that (unless human behaviour changes) global temperatures will rise, but the mathematical models produce a wide range of possible outcomes depending on how you set the assumptions. Despite overall warming, winters in some countries might become colder. Harvests may increase or decrease. The overall impact could be relatively benign or catastrophic. We can guess, we can use our judgement, but we can’t be certain.
In 1952, the science-fiction author Raymond Bradbury wrote a short story called ‘A Sound of Thunder’ in which a time-traveller transported back to the time of the dinosaurs accidentally kills a tiny butterfly, and this apparently innocuous incident has knock-on effects that turn out to have changed the modern world they return to. A couple of decades later, the mathematician Edward Lorenz is thought to have been referencing this story when he coined the phrase ‘the butterfly effect’ as a way to describe the unpredictable and potentially massive impact that small changes in the starting situation can have on what follows.
These butterfly effects are everywhere, and they make confident long-term predictions of any kind of climate change (including political and economic climate) extremely difficult.

MAD COWS AND MAD FORECASTS
In 1995, Stephen Churchill, a 19-year-old from Wiltshire, became the first person to die from Variant Creutzfeldt–Jakob disease (or vCJD). This horrific illness, a rapidly progressing degeneration of the brain, was related to BSE, more commonly known as ‘Mad Cow Disease’, and caused by eating contaminated beef.
As more victims of vCJD emerged over the following months, health scientists began to make forecasts about how big this epidemic would become. At a minimum, they reckoned there would be at least 100 victims. But, at worst, they predicted as many as 500,000 might die – a number of truly nightmare proportions.8 (#litres_trial_promo)
Nearly 25 years on, we are now able to see how the forecasters did. The good news is that their prediction was right – the number of victims was indeed between 100 and 500,000. But this is hardly surprising, given how far apart the goalposts were.
The actual number believed to have died from vCJD is about 250, towards the very bottom end of the forecasts, and about 2,000 times smaller than the upper bound of the prediction.
But why was the predicted range so massive? The reason is that, when the disease was first identified, scientists could make a reasonable guess as to how many people might have eaten contaminated burgers, but they had no idea what proportion of the public was vulnerable to the damaged proteins (known as prions). Nor did they know how long the incubation period was. The worst-case scenario was that the disease would ultimately affect everyone exposed to it – and that we hadn’t seen the full effect because it might be 10 years before the first symptoms appeared. The reality turned out to be that most people were resistant, even if they were carrying the damaged prion.
It’s an interesting case study in how statistical forecasts are only as good as their weakest input. You might know certain details precisely (such as the number of cows diagnosed with BSE), but if the rate of infection could be anywhere between 0.01% and 100%, your predictions will be no more accurate than that factor of 10,000.
At least nobody (that I’m aware of) attempted to predict a number of victims to more than one significant figure. Even a prediction of ‘370,000’ would have implied a degree of accuracy that was wholly unjustified by the data.

DOES THIS NUMBER MAKE SENSE?
One of the most important skills that back-of-envelope maths can give you is the ability to answer the question: ‘Does this number make sense?’ In this case, the back of the envelope and the calculator can operate in harmony: the calculator does the donkey work in producing a numerical answer, and the back of the envelope is used to check that the number makes logical sense, and wasn’t the result of, say, a slip of the finger and pressing the wrong button.
We are inundated with numbers all the time; in particular, financial calculations, offers, and statistics that are being used to influence our opinions or decisions. The assumption is that we will take these figures at face value, and to a large extent we have to. A politician arguing the case for closing a hospital isn’t going to pause while a journalist works through the numbers, though I would be pleased if more journalists were prepared to do this.
Often it is only after the event that the spurious nature of a statistic emerges.
In 2010, the Conservative Party were in opposition, and wanted to highlight social inequalities that had been created by the policies of the Labour government then in power. In a report called ‘Labour’s Two Nations’, they claimed that in Britain’s most deprived areas ‘54% of girls are likely to fall pregnant before the age of 18’. Perhaps this figure was allowed to slip through because the Conservative policy makers wanted it to be true: if half of the girls on these housing estates really were getting pregnant before leaving school, it painted what they felt was a shocking picture of social breakdown in inner-city Britain.
The truth turned out to be far less dramatic. Somebody had stuck the decimal point in the wrong place. Elsewhere in the report, the correct statistic was quoted, that 54.32 out of every 1,000 women aged 15 to 17 in the 10 most deprived areas had fallen pregnant. Fifty-four out of 1,000 is 5.4%, not 54%. Perhaps it was the spurious precision of the 54.32’ figure that had confused the report writers.
Other questionable numbers require a little more thought. The National Survey of Sexual Attitudes has been published every 10 years since 1990. It gives an overview of sexual behaviour across Britain.
One statistic that often draws attention when the report is published is the number of sexual partners that the average man and woman has had in their lifetime.
The figures in the first three reports were as follows:

The figures appear quite revealing, with a surge in the number of partners during the 1990s, while the early 2000s saw a slight decline for men and an increase for women.
But there is something odd about these numbers. When sexual activity happens between two opposite-sex people, the overall ‘tally’ for all men and women increases by one. Some people will be far more promiscuous than others, but across the whole population, it is an incontravertible fact of life that the total number of male partners for women will be the same as the number of women partners for men. In other words, the two averages ought to be the same.
There are ways you can attempt to explain the difference. For example, perhaps the survey is not truly representative – maybe there is a large group of men who have sex with a small group of women that are not covered in the survey.
However, there is a more likely explanation, which is that somebody is lying. The researchers are relying on individuals’ honesty – and memory – to get these statistics, with no way of checking if the numbers are right.
What appears to be happening is that either men are exaggerating, or women are understating, their experience. Possibly both. Or it might just be that the experience was more memorable for the men than for the women. But whatever the explanation, we have some authentic-looking numbers here that under scrutiny don’t add up.

THE CASE FOR BACK-OF-ENVELOPE THINKING
I hope this opening section has demonstrated why, in many situations, quoting a number to more than one or two significant figures is misleading, and can even lull us into a false sense of certainty. Why? Because a number quoted to that precision implies that it is accurate; in other words, that the ‘true’ answer will be very close to that. Calculators and spreadsheets have taken much of the pain out of calculation, but they have also created the illusion that any numerical problem has an answer that can be quoted to several decimal places.
There are, of course, situations where it is important to know a number to more than three significant figures. Here are a few of them:

In financial accounts and reports. If a company has made a profit of £2,407,884, there will be some people for whom that £884 at the end is important.
When trying to detect small changes. Astronomers looking to see if a remote object in the sky has shifted in orbit might find useful information in the tenth significant figure, or even more.
Similarly in the high end of physics there are quantities linked to the atom that are known to at least 10 significant figures.
For precision measurements such as those involved in GPS, which is identifying the location of your car or your destination, and where the fifth significant figure might mean the difference between pulling up outside your friend’s house and driving into a pond.
But take a look at the numbers quoted in the news – they might be in a government announcement, a sports report or a business forecast – and you’ll find remarkably few numbers where there is any value in knowing them to four or more significant figures.
And if we’re mainly dealing with numbers with so few significant figures, the calculations we need to make to find those numbers are going to be simpler. So simple, indeed, that we ought to be able to do most of them on the back of an envelope or even, with practice, in our heads.

2
TOOLS OF THE TRADE
THE ESSENTIAL TOOLS OF ESTIMATION
For most back-of-envelope calculations, the tools of the trade are quite basic.
The first vital tool is the ability to round numbers to one or more significant figures.
The next three tools are ones that require exact answers:

Basic arithmetic (which is built around mental addition, subtraction and a reasonable fluency with times tables up to 10).
The ability to work with percentages and fractions.
Calculating using powers of 10 (10, 100, 1,000 and so on) and hence being able to work out ‘orders of magnitude’; in other words, knowing if the answer is going to be in the hundreds, thousands or millions, for example.
And finally, it is handy to have at your fingertips a few key number facts, such as distances and populations, that crop up in many common calculations.
This section will arm you with a few tips that will help you with your back-of-envelope calculations later on – including a technique you may not have come across that I call Zequals, and how to use it.

ARE YOU AN ARITHMETICIAN?
In the opening section there was a quick arithmetic warm-up. It was a chance to find out to what extent you are an Arithmetician.
Arithmetician is not a word you hear very often.
In past centuries it was a much more familiar term. Here, for example, is a line from Shakespeare’s Othello: ‘Forsooth, a great arithmetician, one Michael Cassio, a Florentine.’ That line is spoken by Iago, the villain of the play, who is angry that he has been passed over for the job of lieutenant by a man called Cassio. It is an amusing coincidence that Shakespeare’s arithmetician Cassio has a name very similar to Casio, the UK’s leading brand of electronic calculator.
Iago scoffs that Cassio might be good with numbers, but he has no practical understanding of the real world. (This rather harsh stereotype of mathematical people as being abstract thinkers who are out of touch with reality is one that lives on today.)
Shakespeare never used the word ‘mathematician’ in any of his plays, though in Tudor times the two words were often used interchangeably, just as ‘maths’ and ‘arithmetic’ are today – much to the annoyance of many mathematicians.
So what is the difference between maths and arithmetic? If you ask mathematicians this question, they come up with many different answers. Things like ‘being able to logically prove what is true’ and ‘seeing patterns and connections’. What they never say is: ‘knowing your times tables’ or ‘adding up the bill’.
Arithmetic, on the other hand, is entirely about calculations.
Here’s an example to show what I mean:
Pick any whole number (789, say). Now double it and add one. By using a logical proof, a mathematician can say with absolute certainty that the answer will be an odd number, even if they are unable to work out the answer to ‘what is twice 789 add one?’1 (#litres_trial_promo)
On the other hand, an arithmetician can quickly and competently work out that (789 × 2) + 1 = 1,579, without needing a calculator.
The strongest arithmeticians can do much harder calculations, too. They can quickly work out in their head what 4/7 is as a percentage; can multiply 43 × 29 to get the exact answer; and can quickly figure out that in a limited-overs cricket match, if England require 171 runs in 31 overs they’ll need to score at a bit more than five and a half runs per over.
My mother, who left school at 17, was a strong arithmetician, as were many in her generation. That was almost inevitable. A large part of her schooling had been daily practice filling notebooks with page after page of arithmetical exercises. But she knew little about algebra, geometry or doing a formal proof, in the same way that many top mathematicians are hopeless at arithmetic.
There is, however, a huge amount of overlap between arithmetic and mathematics. Many arithmetical techniques and short cuts lead on to deep mathematical ideas, and most of the maths that is studied up until school-leaving age requires an element of arithmetic, even if it’s no more than basic multiplication and addition. Arithmetic and maths are both grounded in logical thinking, and both exploit the ability (and joy) of seeing patterns and connections.
And yet, although arithmetic crops up everywhere, after the age of 16 it is very rarely studied. Almost without exception, public exams beyond 16 allow the use of a calculator, and most people’s arithmetical skills inevitably waste away after GCSE.
A while ago, a friend who runs an engineering company was talking with some final-year engineering undergraduates about a design problem he was working on. ‘We have this pipe that has a cross-sectional area of 4.2 square metres,’ he said, ‘and the water is flowing through at about 2 metres per second, so how much water is flowing through the pipe per second?’ In other words, he was asking them what 4.2 × 2 equals. He was assuming that these bright, numerate students would come back instantly with ‘8.4’ or (since this was only a rough-and-ready estimate) ‘about 8’. To his dismay, all of them took out their calculators.
Calculators have removed the need for us to do difficult arithmetic. And it’s certainly not essential for you to be a strong arithmetician to be able to make good estimates. But it helps.

TEST YOURSELF
Can you quickly estimate the answer to each of these 10 calculations? If you get within (say) 5% of the right answer, you are already a decent estimator. And if you are able to work out exactly the right answers to most of them in your head, that’s a bonus, and you can call yourself an arithmetician.
(a) A meal costs £7.23. You pay £10 in cash. How much change do you get?
(b) Mahatma Gandhi was born in October 1869 and died in January 1948. On his last birthday, how old was he?
(c) A newsagent sells 800 chocolate bars at 70p each. What are his takings?
(d) Kate’s salary is £28,000. Her company gives her a 3% pay rise. What is her new salary?
(e) You drive 144 miles and use 4.5 gallons of petrol. What is your petrol consumption in miles per gallon?
(f) Three customers get a restaurant bill for £86.40. How much does each customer owe?
(g) What is 16% of 25?
(h) In an exam you get 38 marks out of a possible 70. What is that, to the nearest whole percentage?
(i) Calculate 678 × 9.
(j) What is the square root of 810,005 (to the nearest whole number)?
Solutions (#litres_trial_promo)

BASIC ARITHMETIC
ADDITION AND SUBTRACTION
The classic written methods for arithmetic start at the right-hand (usually the units) column and work to the left. But when it comes to the sort of speedy calculations that are part of back-of-envelope thinking, it generally pays to work from the left instead.
For example, take the sum: 349 + 257.
You were probably taught to work it out starting from the units column at the right. The first step would be:


9 + 7 = 16, write down the 6 and ‘carry’ the 1.2 (#litres_trial_promo)
You then continue working leftwards:


4 + 5 + 1 = 10, write down the 0 and ‘carry’ the 1; 3 + 2 + 1 = 6.
Working this out mentally, however, it is generally more helpful to start with the most significant digits (i.e. the ones on the left) first.
So the calculation 349 + 257 starts with 300 + 200 = 500, then add 40 + 50 = 90, and finally add 7 + 9 = 16. The advantage of working from the left is that the very first step gives you a reasonable estimate of what the answer is going to be (‘it’s going to be 500 or so …’).
A similar idea applies to subtraction. Using the standard written method, working from the right, 742 – 258 requires some ‘borrowing’ (maybe you used different language). Here’s the method my children learned at school:

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