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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:

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:

8 from 2 can’t be done, borrow 10, 12 – 8 = 4,
5 from 3 can’t be done, borrow 10, 13 – 5 = 8, 2 from 6 = 4.

Starting from the left, however, you can read it as 700 – 200 (= 500), then 40 – 50 (so subtract 10 from 500) and, if you want the exact answer, calculate the units 2 – 8 (subtract 6).

MULTIPLICATION AND TIMES TABLES

Calculators may be here to stay, but children are still expected to learn their times tables in primary school, just as they were one hundred years ago.

In the UK, this means learning all multiplications up to 12. In some countries, such as India, it’s not uncommon for this to be pushed to 20, so that some children might, for example, learn the answer to 13 × 17 off by heart.

When it comes to back-of-envelope maths, knowing your tables up to 10 is generally enough.

You may be a little rusty on your times tables. I’m guessing that the very fact that you are reading this means you can probably calculate 3 × 4 in your head, but many adults are out of practice when it comes to some of the harder multiplications. Notorious for tripping people up is 7 × 8, though according to one analysis of over a million calculations using times table that were done online,3 it is 9 × 3 that is answered incorrectly the most often.

Here are a few tips that can be handy when doing multiplications in your head. These apply to the times tables, but are also handy for multiplying larger numbers.

Tip 1

The order of multiplication makes no difference to the answer. For example, 3 × 5 is the same as 5 × 3. One way to convince yourself why this is true is to think of multiplication as counting eggs in a tray.

How many eggs are in the tray above? Three rows of five, or five columns of three? Either way, it comes to 15. What’s powerful about this truism is that you can think of any multiplication as being like counting eggs in a tray. It means you can be confident that 7,431 × 278 is the same as 278 × 7,431, even if you don’t know what the answer is.

This idea is not just restricted to multiplying two numbers together: 5 × 13 × 2 is the same as 2 × 5 × 13. By rearranging the order of the numbers that you are multiplying, you can often make a calculation easier. In this case, since 2 × 5 = 10, we can arrange 5 × 13 × 2 to become 10 × 13 = 130.

Tip 2

Multiplying by 3 is the same as doubling a number, then adding the number again. Thus, 3 × 12 is the same as 2 × 12 (24), then add another 12.

Tip 3

Multiplying by four is the same as doubling, then doubling again. And to multiply a number by eight, you double it three times.

Tip 4

Instead of multiplying by nine, you can simply multiply your starting number by 10 and then subtract the starting number. For example, 9 × 8 is the same as 10 × 8 (= 80), minus 8 (= 72). Likewise, 9 × 68 is the same as 10 × 68 (680) minus 68 (= 612).

Tip 5

Multiplying by 5 is the same as halving the answer and then multiplying by 10. So, 468 × 5 looks hard. But it is the same as 468 ÷ 2 (= 234) × 10 (= 2,340), which is considerably easier. You can, of course, reverse the order and multiply by 10, then divide by two: 43 × 5 = 430 ÷ 2 = 215.

TEST YOURSELF

Trying working these out in your head (the short cuts mentioned above might help, or use your own method):

(a) 3 × 26

(b) 35 × 9

(c) 4 × 171

(d) 5 × 462

(e) 1,414 ÷ 5

Solutions

DIVISION

Division can be described in many ways, but one way to think of it is simply as the reverse of multiplication … working your times tables backwards. To divide 72 by 8, you mentally check what multiple of 8 gives 72 in the times table (answer 9). More often than not, there will be a remainder, but the idea is the same. So, 74 ÷ 8? The nearest multiple of 8 below 74 is 9 × 8 = 72, so the answer is 9 remainder 2. That’s another reason why fluency with times tables is useful.

Dividing into larger numbers can be done using short division, which is just repeated working out using tables. To work out 596 divided by 4, the script I was taught goes like this:

Four into five goes once, remainder 1 (write 1 on the top; 1 is carried to make 19), four into 19 goes four times remainder 3, four into 36 goes exactly nine times.

You may wonder where this precise method belongs in a book about estimation. The point is that there is no need to follow the calculation through to the end – you can round the answer at any stage. For example, we could have stopped after the second division to get the answer 150 (rounded to two significant figures). Short division is a useful aid for calculating percentages, as we’ll see here.

DECIMALS AND FRACTIONS

PLACE VALUE AND DECIMAL POINTS

For some people, numbers become more difficult when they are smaller than one.

The numbers to the right of the decimal point work in just the same way as those to the left. The first digit after the decimal point is the number of ‘tenths’, the next is ‘hundredths’, then ‘thousandths’ and so on.

In the number 0. 5 2 8 there are five tenths and two hundredths, but you are allowed to read across the columns if you want, so you can also say there are 52.8 hundredths. Another way to write this is 52.8 ÷ 100, or 52.8 ‘per cent’. There’s more about per cents below.

There may be times in everyday life when you need to convert fractions to decimals (or to percentages). A newspaper item might say: ‘1 in 4 have suffered some sort of theft, while 8% have experienced a burglary’. That’s 25% + 8% = 33%: about one-third.

For many fractions, the decimal equivalent is familiar:

1/2 is the same as 0.5

1/4 is 0.25

1/3 is 0.33

But what about two-sevenths (which is 2 ÷ 7)?

One way to convert a fraction to a decimal is with short division – in exactly the same way as you’d work out 200 ÷ 7, but with a decimal point inserted, since you’ll be dealing with numbers smaller than 1:

As a decimal, two-sevenths begins 0.2857 … which you can round to 0.286, or 0.29 etc., depending on the level of precision that you want.

DECIMAL POINTS – A MATTER OF LIFE OR DEATH?

A child weighing 20 kg has an infection and needs a course of the antibiotic amoxicillin. The guideline is to administer 25 mg of amoxicillin per kg of body weight every 12 hours. The medication comes in a suspension of 250 mg per 5 ml. What dose (in ml) should the child be given?

That might sound like the GCSE maths question from hell, but it is in fact a fairly routine problem that might be encountered by a house doctor or senior nurse working on a hospital ward. Have a go at working out the answer, and then imagine what it’s like committing yourself to writing down the dose, knowing that if you put the decimal point in the wrong place, the consequences could be life-threatening.

A calculator might help here, of course, but only if you know which numbers to divide into which other ones – and be careful that fat fingers don’t press the wrong digit, or an extra zero by mistake.

It shouldn’t be a surprise that GPs and hospital workers do sometimes make mistakes with calculations like this. A doctor told me (on promise of being kept anonymous) that on one occasion, he prescribed a drug to a patient, and noticed that a couple of days later the patient’s condition had, if anything, got slightly worse. Wondering why the drug wasn’t working, he checked back and discovered he’d got the dose wrong by a factor of 10. Fortunately, he was giving the patient 10 times too little.

MULTIPLYING FRACTIONS

It’s not often that you’ll have a need to multiply fractions together. By far the most common reason I ever have to do this is when working out probabilities of two events happening (for example, what’s the chance that the next card turned over in poker will be a Queen and the one after that will be another Queen?).

The simple rule for multiplying two fractions is to multiply the two top numbers (numerators) together to make the new numerator, and do the same with the bottom numbers (denominators) to make the new denominator.

For example:

You can simplify the calculation if any of the numbers at the top and bottom of the fractions have a ‘factor’ (i.e. a number that divides into them) in common. For example, this …

… looks difficult. But the 6 on top and the 15 on the bottom are both divisible by 3, so we can simplify it to:

How big is 8/35? Well, 8 ÷ 32 is a quarter, so 8 ÷ 35 is a bit less than a quarter.

TEST YOURSELF

(a)

(b)

(c)

(d) Is bigger or smaller than ?

(e) Work out

Solutions

PERCENTAGES

It’s handy to remember that ‘per cent’ simply means ‘divided by 100’ and the ‘of’ in ‘percentage of … ’ can be translated as ‘and multiplied by … ’ In other words, 30% of 40 is the same as ‘30 divided by 100, and then multiplied by 40’.

This means that in any calculation in the style ‘Find A per cent of B’ (for example, ‘Find 30% of 40’), the answer is going to be A times B divided by 100.

30% of 40?

30 × 40 = 1,200 … divide by 100 … 12.

9% of 80?

9 × 80 = 720 … divide by 100 … 7.2.

Here are some other tips for working out percentages:

Tip 1

Working out 10% of something is easy, so use that as a base. For example, what is 5% of 320? Ten per cent (one-tenth) of 320 is 32, so 5% will be half of that, which is 16.

Tip 2

If 10% doesn’t get you quickly to your answer, try using 1% instead, and multiply up from there. For example, what is 3% of 80? One per cent of 80 is 0.8, so 3% is three times that, which is 3 × 0.8 = 2.4.

Tip 3

In calculations requiring you to work out the ‘percentage of … ’, you can switch the order of the numbers, just as you can in any multiplication. Sixteen per cent of 25 is the same as 25% of 16. And 25% of 16 is the same as saying ‘one-quarter of 16’. In other words, 16% of 25 = 4.

Tip 4

If you are confident with short division (see here), you can quickly become adept at working out percentages to two significant figures in your head – which is as precise as you are ever likely to need a percentage estimate.4

For example, if you score 57 out of 80 in a test, what’s that as a percentage? We can work it out as follows:

57 ÷ 80 = 5.7 ÷ 8.

Now do this as short division:

So, it’s 0.712, which is 71%, to two significant figures (or ‘roughly 70%’).

TEST YOURSELF

(a) A shirt is marked as costing £28, but the shop is offering ‘25% off all marked prices’. What will the new price be?

(b) Work out 15% of 80.

(c) What is 14% of 50?

(d) Estimate 49 out of 68 as a percentage.

(e) Work out 266 ÷ 600 as a percentage to two significant figures.

(f) Kate’s salary is £25,000 and she gets a promotion and an 8.4% pay rise. What is her new salary?

Solutions

REMOVING VAT

A percentage calculation that notoriously catches people out is removing the VAT from a price. At the time of writing, VAT in the UK is 20%. If a price is advertised as being £30 + VAT, then the total price can be worked out either by calculating 20% of £30 (= £6) and adding it to get £36 – or, more directly, by multiplying the original price by 1.20:

£30 (excl. VAT) × 1.20 = £36 (inc. VAT).

So if the price of something is £36 including VAT, does that mean we can remove the VAT simply by taking off 20%? No! Twenty per cent of £36 is £7.20, which means the price without VAT would be:

£36 (inc. VAT) – £7.20 = £28.80.

This is wrong! We know that the price without VAT was £30. What has happened?

To work out the price without VAT, you need to reverse what you did when you added VAT. To add VAT you multiply by 1.2, so to remove VAT you divide by 1.2:

£36 (inc. VAT) ÷ 1.2 = £30 (excl. VAT).

Incidentally, dividing by 1.2 is equivalent to multiplying by 5/6, and a quick way to work out the VAT element in a price that includes VAT (at 20%) is to divide the price by six. So, if the price of an item is £66 including VAT, the VAT element is £66 ÷ 6 = £11. The price without VAT is 5 × £11 = £55. (When VAT was 17.5% during the 1990s, there was no such simple short cut!)

CALCULATING WITH POWERS OF TEN

MULTIPLICATION

Knowing 7 × 8 is one thing, but what about 70 × 80, or 7,000 × 800? Many back-of-envelope calculations involve numbers in the hundreds, thousands and beyond, so it’s important to be able to manipulate these numbers with ease.

Can you work out 700 × 80 without a calculator? A mental short cut that I have always used for a calculation with whole numbers such as 700 × 80 is to treat the leading digits and the zeroes separately. First multiply 7 × 8 (= 56), and then count up the zeroes at the end of the two numbers (there are three altogether) and stick them on the end. So the answer is 56,000.

I’ve asked hundreds of British teenagers to work out 700 × 80, though typically I posed it as a money question: ‘A newsagent sells 700 chocolate bars at 80p each. What is the shop’s total revenue?’

The reassuring thing is that the vast majority of teenagers know that 7 × 8 = 56, several years after their drilling in primary school. Where many struggle is in knowing how many zeroes to put into the answer, and where to stick the decimal point. In answer to the chocolate bar challenge, a significant minority will answer £56 or £5.60 or £5,600, or £56,000. And if teenagers struggle with this (including those who have already passed GCSE maths) it’s safe to assume that many adults do too.

By the way, this is just the sort of problem where estimation can help: 80p is close to £1, and 700 × £1 = £700, so the correct answer is going to be a bit less than £700, and it certainly won’t be £56 or £5,600.

TEST YOURSELF

(a) 400 × 90

(b) 300 × 700

(c) 80,000 × 1,100

(d) Bristol Old Vic Theatre needed to raise money for a complete refurbishment of the building. To help fund-raising, they created 50 ‘silver’ tickets priced at £50,000 each, that would give the purchaser the right to every performance in the theatre in perpetuity. How far did this go towards their ultimate target of £25 million?

Solutions

Multiplication of decimals can be more fiddly. If one of the numbers has zeroes and the other is a decimal, you can ‘trade’ zeroes from one number to the other to make the calculation more manageable. In other words, you multiply one of the numbers by 10 and divide the other by 10, and keep doing this until at least one of the numbers you are multiplying is easy to deal with.

For example:

8,000 × 0.02

= 800 × 0.2

= 80 × 2

= 160.

Or:

0.2 × 0.4 = 2 × 0.04 = 0.08.

DIVIDING BY LARGE NUMBERS

For division, the easiest method is to cancel out zeroes (i.e. keep dividing by 10) on both sides of the equation until you are just dividing by a single digit. So, for example:

12,000 ÷ 40 becomes 1,200 ÷ 4 = 300

And:

88,000 ÷ 300 becomes 880 ÷3 = a bit less than 300.

When dividing by decimals, you can multiply both numbers by 10 until you are no longer dividing by a decimal. For example:

0.006 ÷ 0.02

= 0.06 ÷ 0.2

= 0.6 ÷ 2

= 0.3

TEST YOURSELF

(a) 1,000 ÷ 20

(b) 6,300 ÷ 90

(c) 160,000,000 ÷ 80

(d) 2,200 × 0.03

(e) 0.05 ÷ 0.001

Solutions

USING ‘STANDARD FORM’ FOR LARGE NUMBERS

Scientists often find themselves measuring vast or tiny quantities, and when calculating with these numbers they prefer to use what’s known as standard form. This means expressing all numbers as a single digit followed by the appropriate power of 10. For example, 400 in standard form is 4 × 102.

The powers of 10 are then added (for multiplication) and subtracted (for division).

For example:

90,000 × 40

= 9 × 104 × 4 × 101

= 36 × 105 (or 3,600,000).

and

700,000 ÷ 200

= 7 × 105 ÷ 2 × 102

= 3.5 × 103

TEST YOURSELF

(a) What is 4 × 107 when written out in full?

(b) What is 1,270 written in standard form?

(c) What is 6 billion written in standard form?

(d) (2 × 108) × (1.2 × 103)

(e) (4 × 107) ÷ (8 × 102)

(f) (7 × 104) ÷ (2 × 10−3)

Solutions

STAR WARS POWER

There’s a ‘standard form’ joke that is told about Ronald Reagan’s Strategic Defense Initiative (SDI) of the mid-1980s. I’d love to believe that it really happened.

The idea of the SDI, which was given the nickname ‘Star Wars’, was to develop laser weapons that would be capable of destroying enemy nuclear missiles at long range. The laser weapons would need a huge amount of energy, and millions of dollars were allocated towards researching the feasibility.

At one point during the research, the labs were asked to report back to the government.

‘How much power will one of these weapons need?’ asked one official.

‘We’re going to need 1012 watts, sir.’

‘And how much can you deliver now?’

‘About 106 watts, sir.’

‘OK, good,’ said the government official, ‘so we’re about halfway.’

In case you missed it, ‘halfway’ to 1012 would actually be 5 × 1011, so the official was out by a factor of 500,000.

KNOW YOUR MEGAS FROM YOUR TERAS

The powers of 10 have been given official prefixes that you’ll often encounter in discussions about energy and computer power in particular. Here’s the background to their names..

SI Prefix Origin
103 (1,000) Thousand Kilo From chilioi, meaning ‘thousand’, introduced by the French in 1799.
106 (1,000,000) Million Mega From megas, meaning big or tall. First used as a prefix in late Victorian times.
109 Billion Giga From gigas, meaning giant. Officially adopted in 1960.
1012 Trillion Tera From teras, meaning ‘monster’. ‘Tera’ is similar to the Greek tetra (four) and, coincidentally, this is the fourth prefix.
1015 Quadrillion Peta Adopted in the 1970s, this fifth prefix is a made-up word that is a nod to the Greek number penta but with a consonant left out, copying the pattern from tera.
1018 Quintillion Exa Adopted at the same time as peta, this is hexa with the h removed.
1021 Sextillion Zetta, These prefixes aren’t in common use yet, but as computer power grows, you’ll encounter them more often.
1024 Septillion Yotta

KEY FACTS

To be equipped to do back-of-envelope calculations, there are a few basic statistics that are valuable foundations from which you can build your estimates. Here are some important ones:

World population Between 7 and 8 billion
UK population Approaching 70 million
Distance London to Edinburgh (as the crow flies) 330 miles
Circumference of the equator Around 24,000 miles
Walking speed of a commuter 3–4 mph (a bit below 2 metres/second)
Fastest that a human can run A bit over 10 metres/second (∼20 mph)
Size of a top Premier League football crowd 60,000 is typical
Cruising speed of a regular passenger jet 500–600 mph
Ceiling height of a typical apartment room 2.5 metres/8 feet
Fuel economy of a typical family saloon 40 miles per gallon
Weight of a litre of water 1 kilogram (exactly!)
Weight of a four-seater family car A bit more than 1 tonne, less than 1.5 tonnes
TEST YOURSELF

Using the key facts above as a baseline, you can start estimating other things. Have a go at these:

(a) How far is it from London to Auckland in New Zealand?

(b) How far is it from London to New York?

(c) How many people live in Mexico City?

(d) How tall is a 20-storey office building?

(e) How long would it take a healthy adult to walk 10 miles?

(f) How many children attend primary school in the UK?

(g) How many people get married each year in the UK?

(h) What is the area of the Atlantic Ocean?

Solutions

ZEQUALS

If you have mastered all the other tips in this section, you are now well equipped to do a wide range of rapid estimations without needing a calculator.

There’s one final tool to add: Zequals.

One of the secrets to doing quick back-of-envelope calculations is being able to make the calculations as simple as possible. There are many approaches to estimating, but Zequals is one of the most ruthless and is designed to minimise your need for a calculator. I named it Zequals, and because it has strict rules, I invented a symbol for it, too.

The idea behind Zequals is to simplify all the numbers you are dealing with before you do any calculations, by rounding them to one significant figure. In other words, you are rounding numbers to the nearest unit, or the nearest ten, hundred or thousand – always, and without exception.

The symbol for Zequals is h. Here are some examples of how Zequals operates. Notice how in each case, the number you start with is being rounded so that it only has one digit that is not zero:

6.3 h 6

35 h 40

(the Zequals convention is that if the second digit is 5, you round upwards)

23.4 h 20

870 h 900

1,547,812.3 h 2,000,000 (two million)

Single-digit numbers and numbers with only one non-zero digit stay the same, because they already have one significant figure; so:

7 h 7

0.08 h 0.08

9,000 h 9,000

Why Zequals? The Z stands for zeroes, because this method uses a lot of them. And the zig-zaggy symbol also looks a bit like a saw, which is appropriate, since this technique is a little like brutally sawing off the ends of numbers. And why is Zequals useful? Because it makes complicated calculations so manageable that you can do them in your head, and as we’ll see, it usually takes you to an answer that is in the right ballpark.

Rounding numbers for doing estimations is nothing new, but if you are using Zequals, you have to stick to the rules at all times. And because this is rough-and-ready calculation, you should really ‘zequalise’ the answer too. So:

4 × 8 = 32, but then 32 h 30. So 4 × 8 h 30

A NEW MATHEMATICAL SYMBOL h

The symbol ≈ means ‘is approximately equal to’ and, unusually for maths symbols, there is no hard-and-fast rule for how to use it. For example: 7.3 ≈ 7.2. But it’s also true that 7.3 ≈ 7.0 and that 7.3 ≈ 10.

The approximate value that you choose, be that 7.2, 7.0, 10 or something else, depends on your own judgement of what is most appropriate at the time.

Zequals, on the other hand, has very specific rules. Always and without exception, 7.3 h 7, because it means ‘round the number on the left to one significant figure’.

TEST YOURSELF

What are the following, according to Zequals?

(a) 83 h

(b) 751 h

(c) 0.46 h

(d) 2,947 h

(e) 1 h

(f) 9,477,777 h

Solutions

CALCULATING WITH ZEQUALS

Suppose somebody asks you how many hours there are in a year. Just roughly. There are 365 days in the year and 24 hours in a day, so that’s 365 × 24. That’s hard to work out in your head. But zequalise it and it becomes an easy calculation: 365 × 24 h 400 × 20 = 8,000. Compare that with the exact answer of 8,760. It’s only about 10% off, certainly in the right ballpark for most situations where you’d need to know the answer.

Using Zequals, long division, the bane of many schoolchildren’s lives, becomes a relative doddle.

What’s 5,611 divided by 31?

It zequalises to 6,000 divided by 30, which is 200. Again, that’s not far away from the exact answer (181). Zequals can be useful even when you are using a calculator because you need a more accurate answer. If your calculator is telling you the answer is 18.1, then your estimate using Zequals tells you that the calculator is wrong (most likely because you inadvertently pressed the wrong key at some point).

TEST YOURSELF

Work out the following using Zequals (and you might want to check how close your answer is to the precise answer, and whether your answer is too high or too low):

(a) 7.3 + 2.8 h

(b) 332 – 142 h

(c) 6.6 × 3.3 h

(d) 47 × 1.9 h

(e) 98 ÷ 5.3 h

(f) 17.3 ÷ 4.1 h

Solutions

THE INACCURACY OF ZEQUALS

I can’t emphasise enough that Zequals is not designed to give you exactly the right answer. In fact it can sometimes take you quite a long way from the right answer, particularly because of the Zequals rule that if the second digit is 5 you always round upwards.

Just how inaccurate can it be?

Take the example 35.1 + 85.2. Rounding these to the nearest whole number would give you the answer 120, which is almost exactly right. But according to Zequals, this becomes 40 + 90 = 130, which is nearly 10% too high. What’s more, 130 h 100, which is almost 20% too low.

In multiplication it can be a lot worse.

35 × 65 = 2,275

But according to Zequals: 35 × 65 h 40 × 70 = 2,800, and 2,800 h 3,000. That’s over 30% too high. Does this matter? It depends on what level of accuracy you are looking for.

TEST YOURSELF

(a) Multiplying together any two numbers between 1 and 100, what is the biggest over-estimate you can make if you use Zequals?

(b) And what is the biggest under-estimate?

Solutions

As you’ll find from the Test Yourself box above, if you’re unlucky with the numbers you are dealing with, Zequals can give an answer that is twice or half the right answer – and the more numbers you are putting into a calculation, the greater the scope is for deviating from the target.

At this point, an experienced estimator might want to use a different, more accurate approach. If both numbers are being rounded up to the nearest 10, a common method is to round down one of them to compensate. So, for example, when estimating 35 × 65, instead of calling it 40 × 70, it will be more accurate to round it to 30 × 70. And that is completely sensible if you are looking for a more accurate estimate (and don’t want to use a calculator).

But let’s not forget what the aim is here. What we are looking for is answers that are in the right ballpark. In many situations, ‘the right ballpark’ means ‘the right order of magnitude’; in other words, is the decimal point in the right place? Zequals is all you need in these circumstances – and of course it has the great advantage that it reduces all calculations to being so simple that – with a little practice – you can do them quickly in your head.

And there is perhaps a more important point to make here, which might almost sound like a paradox.

The purpose of Zequals is to make every calculation so simple that almost anybody can do it. And yet knowing when it is appropriate to use Zequals, and knowing how to interpret the results, requires a degree of wisdom and a reasonable confidence with numbers. The better you become at arithmetic, the better you get at using Zequals.

WHO WANTS TO BE A MILLIONAIRE? (PART 1)

September 2001. It was a Celebrity Special edition of Who Wants to be a Millionaire? Jonathan Ross and his wife Jane had made it through to the tenth question. They had £16,000, but had used up their lifelines.

This was the question that would take them to a guaranteed £32,000: ‘Which episode number did Coronation Street reach on 11 March 2001?’

(a) 1,000

(b) 5,000

(c) 10,000

(d) 15,000

The conversation went as follows:

Jonathan: ‘So 50 weeks in a year. Twice a week. 100 a year.’

Jane: ‘There’s not 50 weeks in a year.’

Jonathan: ‘It’s 52 weeks in a year. But roughly. I’m rounding off so the audience can keep up. So it’s about 104 a year. And about 40 years. So that’s … lots. Should we go with (c) 10,000?’

Jane: ‘That’s a ridiculously huge amount … and it can’t be (d).’

Jonathan: ‘It’s (b) or (c), I’ve talked myself out of (d). We’re going to gamble £15,000 of someone else’s money on … (c), 10,000.’

Chris Tarrant: ‘You had £16,000 … you’ve just lost £15,000.’

Jonathan Ross began by doing everything right in his back-of-envelope estimation. His instinct of ‘about 40 years’ of Coronation Street was right. And he was sensible to round the weeks in a year to 50 for simplicity, making it 100 episodes per year, and 100 × 40 = 4,000 would have pointed the couple to the right answer, which was 5,000 (answer (b)). But shifting to the more accurate 52 weeks (making 104 episodes per year) was a distraction, and they never did do that final calculation. It’s an example of where Zequals would have paid off.