Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything

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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

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

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

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: