Maths on the Back of an Envelope: Clever Ways to (Roughly) Calculate Anything

by Rob Eastaway

  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.

  3

  EVERYDAY ESTIMATION

  ESTIMATION AND MONEY

  SHOPPING BILLS AND SPREADSHEETS

  One of the most familiar uses for back-of-envelope maths is in adding up bills. If you’re working to a budget and doing a big shop, it can be useful to mentally tot up how much you’ve put in the basket so far, so as to avoid getting a nasty surprise at the till. But there are also times when you will see a column of figures – in a bill, or in a spreadsheet1 – look at the total and think: ‘surely that can’t be right’. In both cases, speedy ballpark estimates are useful.

  One simple way to speed up the estimate of a shopping bill is to ignore the pence and just add the pounds. The result will be a figure that is an under-estimate – what’s known as a lower bound. You can repeat the task, this time rounding odd pence up to the nearest pound, to give you the upper bound. The true figure will be somewhere in between the two, and a reasonable guess is to go for the middle figure.

  Lower Bound £32

  Upper bound £40

  Estimate £36

  Quicker still, add up the pounds, and then add 50p for each item on the bill, so if the pounds add up to £32 and there are 10 items:

  £32 + 10 × 50p = £37.

  What this doesn’t allow for is the tendency for some products (books, for example) to have a disproportionate number of prices that end with 99p (e.g. £2.99 or £9.99) – or more commonly these days, 95p. This is a device used by retailers to trick us into thinking a product is cheaper than it is. The same items priced at just a penny higher, £3 and £10, feel much dearer, because we are more influenced by the leading digit than the later digits.

  Rather than trying to allow for the peculiar distortions of supermarket prices, you can simplify things considerably just by rounding them to the nearest £1 or by using Zequals.

  TEST YOURSELF

  Bob has a spreadsheet that enables him to keep tabs on sales of widgets around the country. Here’s the column for widget sales in Newcastle:

  SALES

  £ 190.10

  £ 120.46

  £ 8.22

  £ 396.63

  £ 130.50

  £ 41.55

  TOTAL £ 697.36

  Do you trust the total at the bottom?

  Solutions

  HOW CAN THAT SHOP STILL BE IN BUSINESS?

  A few years ago, a new cooking utensil shop opened in my local high street in London. It was at the premium end of the street, where rents had been hiked up extortionately in recent years, and the gossip was that an annual rent of £25,000 was typical for a shop that size. It prompted me to wonder what their business model was.

  If the rent is £25,000 per year, that’s £25,000 ÷ 50 = £500 per week, or about £100 per shopping day. So just to pay the rent, the shop needed to make a profit of £100 per day. If the shop had a 100% mark-up on products, then it needed £200 per day in sales just to cover the rent (in other words, about seven of their upmarket saucepans). That was before the business rates, maintenance and insurance. It meant that before the owners could pay themselves and any other staff, they were probably having to put over £400 through the tills each day, on average. Yet frequently when I went past, there was nobody in the shop. ‘How do they stay in business?’ I wondered.

  My question was answered not long afterwards, when the shop closed.

  SAVING, BORROWING AND PERCENTAGES

  Percentages are ubiquitous when dealing with money. I covered discounts and VAT here, but percentages get more involved when you are dealing with loans and saving – and the numbers when working out what interest you will have to pay on a mortgage, or receive from an ISA, are probably going to dwarf those you encounter during shopping.

  The maths, fortunately, is the same (5% interest on savings of £3,000 means you’ll earn £150 each year). The complication comes with compounding: if your loan or savings run beyond a year, then you start paying, or earning, interest on interest.

  If you have savings of £10,000 in an account that is paying 10% interest (happy days!), you will have £11,000 after one year. But at the end of the second year you won’t have £12,000 because you are now earning interest on £11,000 rather than £10,000. After two years, your savings pot has therefore risen to £10,000 × 1.1 × 1.1 = £12,100. An extra £100. It doesn’t sound much, but this margin becomes more significant the longer you save for, and the higher the interest rate is. (And on the other side, debts that are incurring compound interest can rise alarmingly.)

  When interest rates are small (2.5%, say), there’s a handy back-of-envelope rule for working out longer-term savings that’s close to the right figure. If you are saving for four years at 2.5%, then the interest you’ve earned after four years is very close to 4 × 2.5% = 10%. (For comparison, the correct figure is 10.38%.) The smaller the interest rate, the better this approximation becomes.

  This rule of small numbers allows you to be quite cavalier in calculations where you aren’t required to give a precise answer, because it means you can just add and subtract percentages instead of multiplying and dividing them.

  For example, if your savings go up by 3.3% in one year, 3.1% the next, and 2.7% the year after that, you’re not far off the right number if you say that the total growth over three years will be 3.3 + 3.1 + 2.7 = 9.1% (and using Zequals, you can simplify it further: 3% + 3% + 3% = 9%).

  This is fine over short periods. But over longer periods there’s another handy rule of thumb: the Rule of 72.

  DOUBLE YOUR MONEY – THE RULE OF 72

  If your bank pays you compound interest at 4%, how long will it be before you have doubled your money?

  This complex-sounding calculation can be answered with a deceptively simple rule. It’s known as the Rule of 72.

  Whatever the growth rate (be that 1.2%, 4%, 10% or even 30%), the time it will take for the quantity in question to double can be found by dividing it into 72.

  With an interest rate of 4%, your money in the bank will double in 72 ÷ 4 = 13 years.

  How incredibly convenient (you might be thinking) that the number in this rule of thumb is 72, for 72 is a number that divides exactly by 2, 3, 4 and 6: numbers that will often be used as interest rates.

  It turns out that, strictly speaking, this should be known as the rule of 69.3. That is the figure that emerges from doing the algebra behind exponential growth (described in more detail here). But try dividing anything into 69.3 and you’ll end up with a mess. Whoever first worked out this rule quickly spotted that by nudging it up to 72 instead, there was a chance people would be able to work out the numbers on the back of an envelope – or even in their heads. So the Rule of 72 it is.2

  Knowing how long it will take for numbers to double is handy, but there may be times when you want to know a different target. What about trebling your money, or increasing it tenfold? It turns out there is a rule of thumb for any target of growth that you choose. In each case, it uses a convenient number that is quite close to the accurate one.

  Here’s a table:

  How many years before a quantity … Convenient number to divide into Example: How long it takes if growth rate = 4%

  Doubles 72 72 ÷ 4 = 18 years

  Increases threefold 120 120 ÷ 4 = 30 years

  Increases tenfold 240 240 ÷ 4 = 60 years

  CONVERTING CURRENCY

  Any international trip or purchase is going to involve a conversion, and unlike units of measurement, the conversion rates between sterling, euros
and US dollars are changing all the time. During the current century, £100 could have bought you anything between $120 and over $200, which is a massive variation.

  With many currency conversions, you – or the person you are dealing with – are probably going to be concerned with calculating figures that are correct to the last penny, or cent, and you’ll almost certainly use a calculator. But imagine that you are at the airport and need some US dollars. The exchange rate is at $US1.40 to the pound, you ask the travel-exchange desk for $1,000, and the cashier charges you £793.40. Happy? Well, £793.40 is about £800, and a mental check tells you that 1,000 ÷ 800 = 1.25, which is a long way from the supposed $1.40 rate. So they’re either charging a huge commission, or the cashier has mistyped a number. Do you still want to go ahead with this transaction?

  Fortunately for the British, the pound is more valuable than most of the rest of the world’s units of currency,3 meaning that £1 will usually buy you more than one dollar, euro, Swiss franc, and far more yuan or rupees. Therefore, converting sterling to other currencies generally means multiplying by a number between 1 and 2.

  You probably have your own ideas of mental short cuts you might use to do a rough conversion, but depending on the exchange rate, you might round to the nearest convenient ratio. For example:

  Exchange rate of other currency Close to… Estimation Short cut

  1.09 1.1 Add 10%

  1.35 1.33 Add one-third

  1.52 1.5 Add a half

  1.72 1.75 Add three-quarters

  1.81 1.8 Double, then take off 10%

  2.1 2 Just double it!

  This is fine if you are converting from (say) sterling to US dollars. But if you are doing the reverse, it will mean multiplying by a number less than one, or dividing, and most people find both of these harder to do mentally. A confident arithmetician might be happy dividing by (say) 1.4, but the back-of-envelope approach will be to divide by 2 and add about 50% to the answer. And that’s probably going to be good enough for when you need it. ‘So that hotel is going to cost us $500 for one night – let’s see, halve that = £250, plus £125… more than £350: ouch! – way beyond our budget.’