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

by Rob Eastaway

  ESTIMATING SIZE

  ESTIMATING DISTANCES

  The most straightforward way to estimate distances is by comparing the distance you are trying to figure out to one that you already know. If you know that the distance from Sydney to Melbourne is 500 miles, then the distance from Sydney to Canberra – a city that for historical, political reasons needed to be roughly in the middle of the two cities – is going to be about 250 miles. And if you don’t know the distance from Sydney to Melbourne, you might be able to estimate it using some other information you have – for example, that it takes about an hour to fly from one to the other. Since most planes fly at about 500 mph, in one hour the plane will cover 500 miles. There is, of course, plenty of ‘ish’ involved here.

  The same principle applies to shorter distances, of course. If you know that your own height is 1.5 metres (say), then you can estimate the height of the room you are in by, for example, picturing whether you’d reach the ceiling if you stood on your own shoulders. (I’m a firm believer in picturing things like this.)

  All of this is fairly routine and familiar. But there are three rather more quirky (and I think charming) ways of estimating distance and height.

  1. The Dropped-Stone Method

  When I was a child, we’d often take a trip to Beeston Castle in Cheshire. One of my favourite features there was the well, and we’d delight in dropping a pebble into the well and counting the seconds before we heard it hit the bottom. (With decades-worth of children doing the same thing since, I fear the well is no longer quite as deep as it was.)

  You can estimate the depth of the well with a bit of Newtonian maths, which says that the distance travelled by an object falling under gravity is given by:

  Distance = ½ a.t2

  where a = the acceleration due to gravity (about 10 metres per second per second) and t = the time taken for the stone to drop to the bottom of the well. If the pebble drops for three seconds, the depth of the well is therefore roughly:

  ½ × 10 × 32 = 45 metres.

  There are two things being ignored here that mean the measured time will be an over-estimate. The first is that, because of air resistance, the pebble will eventually stop accelerating, and will therefore take longer to hit the bottom than it would have done in a vacuum. The second is that when the pebble hits the bottom of the well, it takes time for the sound to travel back up to your ears. Both of these effects are quite small, however. So, for a decent estimate of the height, square the time and multiply by 5.

  2. The Finger Method

  Let’s say you are standing on the beach and can see a yacht on the horizon. You wonder how far away the yacht is.

  Here’s one way to find out. Stretch out your arm in front of you, close one eye and hold up a finger so that it covers the yacht.

  Now open that eye and close the other one. Your finger will appear to jump to the side, away from the yacht (a phenomenon known as ‘parallax’).

  The distance to the yacht is roughly:

  10 × the distance that the yacht jumped.

  You will need to use your judgement to estimate by how many yacht-lengths the yacht has jumped to the side. If you estimate that the yacht jumped by about 15 times its own length, then the yacht is roughly:

  10 × 15 yacht-lengths

  = 150 yacht-lengths away from you.

  There is of course one other thing you need to estimate: the length of the yacht itself. You’ll need to use your rudimentary knowledge of yachts to decide if this looks like a 5-, 10- or 20-metre yacht. If you reckon it’s a 10-metre vessel, then your estimate is that it is about 10 × 150 = 1,500 metres away.

  As I say, it’s quirky. But I bet you now want to try it with that pylon you can see out of the window…

  3. The Crisp-Packet Method

  You are in the park and you are curious to know how tall a particularly fine poplar tree is. You can make a good estimate using an empty crisp packet.

  Fold over the corner of the packet so that the top of the packet lines up with the side, and make a crease on the diagonal where the fold is. This diagonal is now at 45 degrees.

  Hold the crisp packet next to your eye, and look along the diagonal as if it’s a telescope, keeping the bottom of the crisp packet horizontal.

  Walk towards the tree until the diagonal of the crisp packet lines up with the top of the tree.

  Now, taking large strides of about one metre, count how many strides it is to the base of the tree.

  The height of the tree is approximately:

  the number of strides + your height.

  How does this work? By lining up the folded crisp packet with the top of the tree, you have formed an isosceles triangle: the distance from you to the tree is the same as the height of the tree above your eye-line.

  CIRCLES AND PI

  There are two formulae related to circles that every child is required to learn.

  For a circle of radius R:

  Circumference = 2πR.

  Area = πR2.

  But what value is π? It depends who you ask.

  A mathematician will tell you that pi is the ratio of the circumference of a circle to its diameter, a transcendental number that begins 3.14159… and continues for ever.

  An engineer (so the joke goes) will tell you ‘π is about 3, but let’s call it 10 just to be on the safe side’.

  Whichever of these views you sympathise with, when it comes to most real-world problem solving, knowing that π h 3 is good enough.4

  But when on earth might you need to use it at all?

  In the 2004 Olympics in Athens, British athlete Kelly Holmes won gold in the 800 metres. Five days later, she had made it to the final of the 1500 metres, and she was aiming to become the first British athlete to win gold medals in both distances.

  Kelly Holmes was a tactical runner, and was prepared to run at a pace that was comfortable for her, even if it meant she spent some of the race at the back of the field. As the final lap began, she was positioned in eighth place. Now she had to get to the front. The problem was that, in overtaking, she would need to be in the second lane and run outside the athletes in front of her. On the straights this would make no difference, but both ends of the track are semicircles, and Holmes therefore had to run round a circle with larger circumference than her competitors. In other words, to win gold, she had to run further than 1,500 metres.

  But how much further?

  At first glance it would appear that we don’t have enough information. How long is an Olympic race track? What’s the radius of the bends? How wide are the bends? But it turns out that only one of these items of data is important.

  Take a look at the sketch of a track. Let’s call the length of the straights L, and the radius of the inside lane at the end R. Remembering that the circumference of a circle is 2πR, the length of a lap is twice the length of the straights plus the circumference of the circle, or 2L + 2πR. But Kelly Holmes had to run around a circle whose radius was larger – by the width of one lane.

  How wide is a lane on an athletics track? Just picture it in your mind. Thirty centimetres? No, much wider than that. A couple of metres (the equivalent of an athlete lying across it)? No, less than that. A metre sounds about right.5 So let’s say the radius of Kelly’s circle was R + 1.

  We can now work out the length of Kelly’s lap:

  2π(R + 1) + 2L

  = 2πR + 2π + 2L.

  Subtract from it the length of the inside lap and Kelly’s ‘extra’ distance is:

  = 2πR + 2π + 2L – 2πR – 2L.

  2πR and 2L cancel out to leave us with 2π…, which is 2 × 3.14… let’s call it 6 metres (since everything is an approximation here).

  Six metres – that’s a lot. It’s the difference between a gold medal and being an also-ran.

  What’s interesting is that Kelly Holmes must have built this into her tactical calculations for the race: she knew she’d have to run further, but felt it was a price worth paying for enabling her to run at her own pace. A
nd it worked: she won the race with a couple of metres to spare.

  And that is how Kelly Holmes became Dame Kelly Holmes.

  AREAS AND SQUARE ROOTS

  We are often presented with numbers that are in square units, particularly area. A description of an apartment might say that it is ‘1,200 square feet’, while a forest fire might be said to be covering ‘100 square miles’. Picturing square anything is difficult – we tend to find it easier to think in lengths. A hundred square miles is the equivalent of a square with sides that are 10 miles long, and to find the length of the side of that square, we need to be able to work out the square root of the area.

  Here’s a real example. In the winter of 2013–14, the South West of England experienced one of its wettest months ever. As a result, an area of low-lying land known as the Somerset Levels was flooded, and the flood waters remained for several weeks. At its peak in January 2014, it was reported that 69 square kilometres had been flooded.

  Let’s picture how big this area of flooding would be if it were a square.

  If the area of a square is 69 km2 then the length of each side is

  which is a number between 8 and 9 (and nearer to 8). So, the area that was flooded was roughly the same as a square that was 8 km × 8 km, or 5 miles × 5 miles. Now that is something that I can just about picture.

  The Somerset Floods story was an example of where it could be handy to be able to work out the square root of a number. Working this out exactly can be messy, but there’s a neat method for making a good estimate.

  Suppose you want to work out the square root of 170,423.

  Starting from the right-hand side of the number (the units column) and working leftwards, break the number up into pairs of digits, like this:

  17 04 23.

  Start with the first pair of digits (17) and estimate the square root of that number. Since the square root of 16 is 4, the square root of 17 is going to be 4-and-a-bit. If we’re using Zequals, we just call it 4, but if we want a little more accuracy we can call it 4.1.

  Now count how many other pairs of digits there are, and for each pair, multiply the square root of the first number by 10. In this case, there are two other pairs, so we multiply 4.1 × 10 × 10 = 410. So the square root of 170,423 is about 410.

  Try another: the square root of 4,138,947.

  Split the number into pairs, starting at the right:

  4 13 8947

  (notice this time that the opening ‘pair’ is just a single digit: 4).

  The square root is therefore, roughly:

  2 × 10 × 10 × 10 = 2,000.

  TEST YOURSELF

  Can you estimate in your head the square roots of the following numbers? If you end up within 5% of the exact answer, give yourself a point. Within 1%, give yourself a hefty pat on the back.

  (a) 26

  (b) 6,872

  (c) 473.86 (hint – ignore the digits after the decimal point!)

  (d) The floor area of a flat is advertised as being ‘910 square feet’. How big would that be if it were a single square room?

  (e) According to Wikipedia, the Caspian Sea is 371,000 km2. If it were a square with the same area, would it fit inside the borders of France?

  Solutions

  WHO WANTS TO BE A MILLIONAIRE? (PART 2)

  The year was 2008,6 and this was an episode of Who Wants to be a Millionaire? featuring couples. One couple – let’s call them the Smiths – had made it to £64,000.

  The question that would take them to £125,000 was this: ‘Which ocean has an area of 4.7 million square miles?’

  (a) Arctic

  (b) Atlantic

  (c) Indian

  (d) Pacific

  The Smiths didn’t know the answer, so decided to use their final lifeline, which was ‘Ask the audience’.

  About half the audience voted for the Pacific, but the couple had hoped for a more definitive vote – 80% or more – so they decided to play safe and take the money.

  The reason I know about this story is that my friend John Haigh, a lecturer at Sussex University and keen back-of-enveloper and Zequaliser, told me the next day that this question had cropped up, and that he had worked out the answer in his head. As his starting point, he made an estimate of the size of the ocean with which he was most familiar: the Atlantic. Can you figure out which was the correct answer? (See here)

  METRIC AND IMPERIAL CONVERSIONS

  WHO NEEDS IMPERIAL?

  Like it or not, we will all continue to need to convert from metric to imperial and vice versa for some time yet. Why? There are two reasons:

  (1) The USA: The world’s biggest economy and most influential culture still stubbornly talks and works in feet, yards, pounds and gallons. This rubs off on the rest of the world because there will be references to imperial units not only in engineering technical specifications but also, for example, in popular songs, movies and American cookery books.

  (2) Old habits die hard: Commonwealth countries that shifted from imperial to metric still have a legacy of imperial measurements in their language and their thinking. For example, in New Zealand, which went fully metric in 1976, the number of kilometres that a car has travelled is still referred to as its ‘mileage’. But that’s nothing compared with the UK, which is divided down the middle as a nation on which system it uses. Even individuals have split personalities when it comes to measurement. I have met numerous adults who, for example, know their height in feet but their weight in kilos. And this is not just an age thing. I’ve done surveys of hundreds of 15-year-olds across the UK, and the results are consistent:

  About 75% of 15-year-olds estimate other people’s height in feet and inches.

  About 30% of them estimate other people’s weight in stones and pounds.

  This widespread use of imperial units is despite the fact teenagers never encounter these units in their school exams, and despite the fact that almost none of them know that there are 14 pounds in a stone! So, in the UK, whichever units of measurement you use, you are going to encounter people who prefer the other units.

  THE MARS ORBITER FIASCO

  In December 1998, NASA launched a space probe called the Mars Climate Orbiter. Its mission: to study the Martian atmosphere. Several months later, as the orbiter approached the planet, it fired the thrusters that were designed to put it into a stable orbit. But to the horror of the NASA team who were monitoring progress, the rocket thrusters were much too strong and the probe hurtled into the planet and was destroyed.

  A NASA review board later discovered that the software designed by the Jet Propulsion Laboratory at NASA had used the metric system in its calculations, but the engineers at Lockheed Martin Astronautics who built the spacecraft had based their calculations on traditional inches and feet (in the report, this was referred to as the ‘English system’, as if this were somehow not the USA’s fault). Instead of applying pound-force, the rockets applied Newton-force, about four times bigger. The cost of this simple error was $125 million of lost space probe.

  MILES AND KILOMETRES

  We took a huge step forward when we switched from imperial units to metric. Calculations in feet, pounds and gallons were simplified overnight when everything could be worked out in decimal.

  In the UK, metrication really kicked in during the early 1970s, when we joined the EEC, in which metric units were already standard. The school curriculum went metric at the same time, which means that maths education of anyone under the age of 50 focused exclusively on metric units. With one, glaring exception: the mile.

  Road signs are exclusively in miles, and, by default, every car speedometer is in miles per hour. We therefore have the curious situation where the vast majority of teenagers will quote long distances in miles, but short distances in metres; and they will quote faster speeds in miles per hour, but slower, more human speeds in metres per second. Confusing? Well no, not really. So long as they aren’t having to switch between the two, most people are comfortable enough working in whichever imperial or metr
ic unit they are used to in the context they are working in.

  The problems start when there is a need to convert from metric to imperial. I asked a large group of 15-year-olds to estimate the distance from London to New York. Their answers varied considerably, but most of them were in the not-completely-outrageous range between 1,000 and 10,000 miles (the correct figure is around 3,500 miles as the crow – or at least the Boeing 787 Dreamliner – flies).

  The problems arose when they were asked to then quote that number in kilometres. There was relatively low awareness of what the relationship is between a mile and a kilometre, other than that they are both ‘quite long’. In many cases, the mileage figure was just multiplied by 10 to get a figure in ‘kilometres’. Perhaps this mistake arises because the shorthand for both metres and miles is ‘m’. There’s a vague awareness that an ‘m’ is related to a ‘km’, so why not multiply by 1,000… no wait, that sounds too high… why not multiply by 10 instead?

  The actual ratio of kilometres to miles is 1.6. (More precisely, it is 1.609…)

  How To Remember Your Conversions

  These three mnemonics appeared on the back of Kellogg’s cornflakes packets in the 1970s, and those of a certain generation have never forgotten them:

  A litre of water’s

  a pint and three-quarters