Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything
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HarperCollinsPublishers 1 London Bridge Street London SE1 9GF
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First published by HarperCollinsPublishers 2019
FIRST EDITION
© Rob Eastaway 2019
Cover design by Andrew Davis © HarperCollinsPublishers Ltd 2019 Cover photograph © 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
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Source ISBN: 9780008324582
Ebook Edition © September 2019 ISBN: 9780008324599
Version 2019-08-30
CONTENTS
1 Cover
2 Title Page
3 Copyright
4 Prologue: How Many Cats?
5 THE PERILS OF PRECISION
6 TOOLS OF THE TRADE
7 EVERYDAY ESTIMATION
8 FIGURING WITH FERMI
9 LAST WORD: WHEN THE ROBOTS TAKE OVER
10 Appendix
11 Answers and Tips
12 Endnotes
13 Acknowledgements
14 About the Author
15 About the Publisher
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1 Cover
2 Frontmatter
3 Begin Reading
4 Table of Contents
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 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
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 many significant figures (this is an important concept – please see here) 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
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: