Maths on the Back of an Envelope: Clever Ways to (Roughly) Calculate Anything
Page 13
A × (1 + R)N = 2A
Cancelling A on both sides:
(1 + R)N = 2
Take logarithms of both sides:
N.ln(1 + R)= ln 2 = 0.69 (= 69%)
There is a rule of thumb familiar to mathematicians that if R is small then ln(1 + R) ≈ R (this is accurate to within 5% if R < 10%). In other words:
N × R = 0.69
N = 69% ÷ R.
This is why it should really be the Rule of 69. The number is adjusted to 72 because 72 is a multiple of many standard interest rates: 1%, 2%, 3%, 4%, 6%, 8% and so on.
WHO WANTS TO BE A MILLIONAIRE? (PART 2)
The ocean is the Arctic. John estimated that the Atlantic is about 30 million square miles, using similar calculations to the one at the bottom of here. That’s getting on for 10 times bigger than the 4.7-metre-square-mile ocean in the question. The Indian Ocean is a similar size to the Atlantic, and the Pacific is bigger than both of them. No doubt the reason most of the audience voted for the Pacific is that 4.7 million is a really big number, and they knew that the Pacific is also really big. But, of course, ‘really big’ and ‘really really big’ are not the same thing.
RANGE OF A SHOT PUT
The range of a shot put can be worked out from this complex-looking formula:
where:
R is the range of the shot put;
υ is the speed of the shot put when released;
g is the acceleration due to gravity;
θ is the angle relative to the horizontal of the shot put when it is released;
y0 is the height above the ground at which the shot put is released;
If gravity is the only thing that changes, then the range increases roughly as I indicated here. In reality, in lower gravity it should be possible to release the shot at a higher speed (the shot will feel less heavy, so you can push it faster). This will increase the range, so my estimate of the advantage of Mexico City is on the low side.
HOW LONG TO COUNT TO A MILLION?
‘Seven’ and ‘Eight’ are the longest words from 0 to 9, and ‘seventy’ is the longest tens word, so the smallest number to exceed 280 characters is going to depend heavily on sevens. 777,777,777,777,777,777 (seven hundred seventy-seven quintillion seven hundred seventy-seven quadrillion seven hundred seventy-seven trillion seven hundred seventy-seven billion seven hundred seventy-seven million seven hundred seventy-seven thousand seven hundred seventy-seven!) uses 255 characters. We have 25 characters left. In front of that number we put the sextillions (sextillion has 11 characters including the space). The smallest number to exceed 25 characters is: one hundred one sextillion. So the Count will be frustrated when he gets to 101,777,777,777,777,777,777.
Phew!
ANSWERS AND TIPS
ENVELOPES VERSUS CALCULATORS
(a) 17 + 8 = 25 The vast majority of adults and teenagers in my survey answered this in their head, but even for a straightforward addition like this, there is a range of ways that people use to get there. The three common approaches were:
7 + 8 = 15, then add 10 to get 25.
Split 8 into 3 + 5; then add 17 + 3 = 20; then add 5 to get 25 (splitting a number up in this way is referred to as partitioning in primary schools).
8 is two less than 10 … 17 + 10 = 27, then take away 2 to get 25.
(b) 62 – 13 = 49 Almost everyone uses two steps here. Those who do it mentally either do ‘take away 10 = 52, then take away three = 49’ or ‘take away three = 59, then take away 10 = 49’. Those who use a written method will typically work from the right: ‘3 from 2 … borrow 10 …, etc.
(c) 2,020 – 1,998 = 22 Viewed as a regular subtraction, 2,020 – 1,998 requires careful carrying over of tens and hundreds. But if the problem had been: ‘Amy was born in 1998. How old will she be in 2020?’ most people solve this by counting up, rather than by subtracting: ‘two years from 1998 to 2000, add 20 years up to 2020, equals 22 years’.
(d) 4 × 9 = 36 Those who do calculations frequently will remember their times tables, and regurgitate ‘four nines are thirty-six’ without having to think. But it’s interesting to observe how those who are rusty with their times tables work it out. The quickest way is to calculate 10 × 4 (= 40) and then subtract 4.
(e) 8 × 7 = 56 Aside from instant recall, approaches that adults shared with me included:
7 × 7 = 49, add 7 = 56
2 × 7 = 14, double it = 28, double that = 56
5 × 7 = 35, add 7, add 7, add 7 = 56.
(f) 40 × 30 = 1,200 People generally know, or very quickly work out, 4 × 3 = 12. But change it to 40 × 30, and the addition of those zeroes can make this calculation much more of a struggle. A common approach is to do it in two steps: reduce one of the numbers to a single digit (e.g. 40 × 3 = 120), then multiply by 10 to give 40 × 30 = 1,200. There are others, however, who just guess. The answer 12,000 is not uncommon. (See also the ‘count the zeroes’ rule here.)
(g) 3.2 × 5 = 16 Of the calculations so far in the quiz, this is the first for which almost everybody uses a written method. Most common is: 5 × 3 = 15; 5 × 0.2 = 1; 15 + 1 = 16.
Some shortcuts for multiplying by 5 can be found here.
(h) 120 ÷ 4 = 30 There are two common strategies for dividing by four. The first is to halve the number and then halve it again (120 ÷ 2 = 60, and 60 ÷ 2 = 30). The other is to do mental short division: ‘four into 12 goes 3, so the answer is 30’.
(i) Three-quarters 75% is such a commonly used quantity that many people are familiar that it is three-quarters without needing any thought. Some figure it out by starting at 25% (one-quarter) and multiplying it by three.
(j) 10% of 94 = 9.4 The most common approach that adults use is to shift the decimal place one to the left, so that the hundreds become tens, tens become units etc.
ARE YOU AN ARITHMETICIAN?
(a) £2.77 change. This sort of problem would usually be regarded as a subtraction, yet somebody working behind a bar would typically treat it as an addition: start with £7.23, round up 7p to £7.30, then add 70p to get £8, then add £2 to get to £10.
(b) Gandhi was 78 when he died. Done as a subtraction (1948 take away 1869) this can get messy, even before getting your head around the months of Gandhi’s birth and death. As with the change example in (a), it’s easier if you treat this as an addition: It’s 31 years from 1869 to 1900, plus 48 years after 1900. 31 + 48 = 79. However, Gandhi died before his October birthday, so he was still 78 years old.
(c) 56,000 pence or £560. The most common error here is getting the number of zeroes wrong. You’ll find tips on placing the zeroes and decimal point here.
(d) Her new salary is £28,840. In surveys of adults, typically only around 25% (that’s one in four) are comfortable doing calculations involving percentages – even when they have a calculator.
(e) 32 miles per gallon. Dividing 144 by 4.5 would be beyond the mental arithmetic capabilities of almost anybody. The ‘trick’ here is knowing how to make the calculation easier. 4.5 is a difficult number – nobody wants to divide by that. But if you double 4.5 you get 9. Instead of 144 ÷ 4.5, you can double top and bottom to turn it into 288 ÷ 9. With mental short division (see here), this is then relatively easy: 9 into 28 goes 3, remainder 1, 9 into 18 goes 2; answer … 32.
(f) £28.80.
(g) 16% of 25 is 4. Most numerate adults confronted with this question break it down into steps: to find 16%, first work out 10%, then 5%, then 1%. Which is fine – it works. But it can be turned into a single step if you realise that 16% of 25 is exactly the same as 25% of 16.
(h) 54%. We’re now reaching the calculations where almost everyone would automatically reach for a calculator. How do you even begin to figure this answer out exactly? Forced to do it mentally, some people spot that 38 ÷ 70 is close to 40 ÷ 70, which is four-sevenths. If you know that 1/7 is roughly 14% (and quite a few people do) then 4/7 is going to be four times that: about 56%. But now to adjust this down – a bit of intelligent tweaking suggests it’s going to be somewhere betw
een 54% and 55%, but which is it? Mental short division (see here) delivers an answer in a few seconds: it’s 54.3%, or 54% to the nearest whole per cent.
(i) 6,102. If you attempt 678 × 9 using long multiplication you’ll probably get in a real tangle, as you try to carry figures over in your head. The short cut here (which you can use whenever multiplying by 9) is to multiply by 10 instead. 678 × 10 = 6,780. Now subtract 678. Answer – 6,102.
(j) 900. Working out the square root of 810,005 precisely is hard – that ‘5’ on the end is a pain. But the square root of 810,000 is much easier. 92 is 81, and 9002 is 810,000. You’ll find a general method for working out square roots here.
MULTIPLICATION AND TIMES TABLES
Here are some ways that you might have got to the answer:
(a) 3 × 20 = 60, plus 3 × 6 = 18, giving the answer 78. Or double 26 (= 52) then add 26 (= 78).
(b) 35 × 10 = 350, subtract 35 to get 315.
(c) Multiplying by 4 is the same as doubling twice: 171 × 2 = 342, and 342 × 2 = 684.
(d) Multiplying by 5 is the same as dividing by 2 and multiplying by 10, hence 462 ÷ 2 = 231, and × 10 = 2,310.
(e) Dividing by 5 is the same as multiplying by 2 and dividing by 10: 1414 × 2 = 2,828, divide by 10 to get 282.8.
MULTIPLYING FRACTIONS
(a) 1/3 × 1/2 = 1/6.
(b) 2/5 × 2/3 = 4/15, just over one-quarter.
(c) 3/4 × 1/5 × 2/3 = 6/60 = 1/10. (You can cancel out the 3s on the top and bottom to simplify the calculation to 1/4 × 1/5 × 2 = 2/20.)
(d) 6/7 × 14/23 = 6 × 2/23 = 12/23, i.e. just over one-half.
(e) To work out 51/52 × 50/51 cancel the two 51s to give you 50/52 (which is about 96%). Here’s a real-world application of this calculation: the chance that the Ace of Spades won’t be the top card or the second card in a shuffled pack of 52 cards is 51/52 × 50/51, which is 50/52.
PERCENTAGES
(a) £21 (25% of £28 is £7).
(b) 12 (10% of 80 is 8, plus 5% (4) = 12).
(c) 7. (Remember that 14% of 50 is the same as 50% of 14.)
(d) About 70%. (49 ÷ 68 is close to 49 ÷ 70, and 4.9 ÷ 7 = 0.7.)
(e) 44%. (Using short division, 2.66 ÷ 6 = 0.44 … and you can stop there.)
(f) New salary £27,100. A short cut here is to spot that 8.4% of 25 is the same as 25% (or one-quarter) of 8.4 = 2.1. So Kate’s salary increase is 2.1 × 1,000 = £2,100. (You’d get a good estimate of the answer by saying that 8.4% is roughly 10%, so her pay rise will be a bit below £2,500.)
MULTIPLICATION
(a) 36,000 (4 × 9 = 36, and three zeroes).
(b) Four zeroes, so 210000 = 210,000.
(c) 88 followed by six zeroes, so 88 million.
(d) 50 × 50,000 = 25 followed by five zeroes, 2,500,000 – which was one-tenth of their target. (A true story, by the way.)
DIVIDING BY LARGE NUMBERS
(a) 100 ÷ 2 = 50.
(b) 630 ÷ 9 = 70.
(c) 2,000,000 (2 million).
(d) The same as 220 × 0.3 = 22 × 3 = 66.
(e) The same as 50 ÷ 1 = 50.
USING STANDARD FORM FOR LARGE NUMBERS
(a) 40,000,000 (40 million).
(b) 1.27 × 103.
(c) 6 × 109 = 6,000,000,000.
(d) 2.4 × 1011.
(e) 0.5 × 105 or, more correctly, 5 × 104.
(f) 3.5 × 107 (which is 35 million).
KEY FACTS
(a) A bit under 12,000 miles. New Zealand isn’t quite halfway around the planet from the UK.
(b) About 3,500 miles. It’s roughly quarter of the way around the earth. Or, if you have ever flown to New York, you will know that the flight takes six or seven hours. The plane will be flying at a bit below 600 mph. So the distance is going to be 600 × 6 = 3,600 miles (ish).
(c) About 9 million. Mexico is one of the world’s bigger cities. So is London. Mexico’s population is going to be nearer ten million than one million.
(d) 200 feet or 60 metres (though it will vary). If we allow 10 feet (3 metres) for each storey, then 200 feet/60 metres is a reasonable estimate.
(e) Three hours? Walking at a brisk 4 mph it would take 2½ hours, but few people can keep up that pace for long distances. Call it three hours.
(f) About five million. Children attend primary school between the ages of 4 and 11 (there are seven year groups, including reception). The very crude sketch below shows the spread of population in more economically developed countries. It’s fairly even across all age groups from 0 to 70, and starts tailing off after that.
If we simplify this even more and imagine the population is evenly distributed across the ages 0 to 80, then in the UK there are roughly:
70 million people ÷ 80 years
~ 900,000 people in each age group.
That suggests around 7 × 900,000 = 6.3 million primary children – six million is a decent rounded estimate (the official figure is around 5 million).
(g) Around 250,000. From the previous question, we’ve got an estimate of around 900,000 people in each age group. Let’s pick 30-year-olds and pretend that’s the age when all people get married. If, say, half of all the population gets married at some point in their life, that makes 450,000 people getting married, and since it takes two to make a marriage, that suggests 225,000 (call it 200,000) weddings. Of course some people get married at 16, others at 60, but this doesn’t change the answer, as long as people only get married once. In reality, some people do have more than one wedding, but this is the minority, so the average is unlikely to be more than, say, 1.2 weddings per person. That suggests around 250,000 weddings per year. (This isn’t far from the official statistics, though the number of weddings is declining.)
(h) Official figures say anything between 30 and 40 million square miles. The Atlantic is a complicated shape. If we want to estimate its size, it’s easier to think of it as a rectangle that fills the gap between Europe/Africa and the Americas. Let’s call the width of the rectangle 3,500 miles (the London–New York distance); see answer (a) above). The Atlantic spans a large part of the globe north/south, so call its height 10,000 miles. The area is therefore roughly 3,500 × 10,000 = 35 million square miles.
ZEQUALS
(a) 83 h 80.
(b) 751 h 800.
(c) 0.46 h 0.5.
(d) 2,947 h 3,000.
(e) 1 h 1.
(f) 9,477,777 h 9,000,000.
CALCULATING WITH ZEQUALS
(a) 7.3 + 2.8 h 7 + 3 = 10.
(b) 332 − 142 h 300 − 100 = 200.
(c) 6.6 × 3.3 h 7 × 3 h 20.
(d) 47 × 1.9 h 50 × 2 = 100.
(e) 98 ÷ 5.3 h 100 ÷ 5 = 20.
(f) 17.3 ÷ 4.1 h 20 ÷ 4 = 5.
INACCURACY OF ZEQUALS
(a) The biggest over-estimate is for 15.1 × 15.1 = 228.01; according to Zequals it is 20 × 20 = 400, which is 75% higher than the correct answer.
(b) The most extreme under-estimate is 14.9 × 14.9 = 222.01. Zequals rounds this to 10 × 10 = 100, which is over 55% too low.
SHOPPING BILLS AND SPREADSHEETS
The total is short by £190.10 – the value at the top of the column has been missed out. If you start by adding the hundreds column, you get £600 … which looks promising. But using Zequals, for example, you end up at £800, more than £100 above the £697.36 total, which should raise your suspicions. Rounding all the numbers to the nearest hundred also gives you £800. This is enough evidence to suggest that something is up – which indeed it is.
AREAS AND SQUARE ROOTS
These answers are correct to 3 significant figures. How close did you get?
(a) 5.10. If you estimated 5, give yourself a point. If you thought 5.1 – two points!
(b) 82.9. Split the number as 68 72. 68 is a bit more than 64, which is 82, so the answer will be a bit more than 8 × 10 = 80. If you estimated anything in the range 82 to 84, give yourself two points.
(c) 21.8. The decimal point is a distraction. 473.86 h 500, so the answer is going to
be a little more than 20. If you estimated 22, give yourself two points.
(d) 30.2. Split the number up as 9 10. This is close to 9 00, so the answer will be a bit more than 3 × 10 = 30, i.e. the room would be roughly 30 foot square (or 10 metres × 10 metres).
(e) 609 km × 609 km (roughly 400 miles square, so it would comfortably fit inside France). Split the number as 37 10 10. 37 is close to 36, so the answer is going to be a little more than 6 × 10 × 10 = 600.
BACK-OF-ENVELOPE CONVERSIONS
(a) Rough: 140 km Accurate: 112 km
(b) Rough: 80 lb Accurate: 90 lb
(c) Rough: 150 yards Accurate: 163 yards
(d) Rough: 50 miles Accurate: 63 miles
(e) Rough: 80 °F Accurate: 77 °F
(f) Rough: 70 kg Accurate: 62 kg
Endnotes
1: THE PERILS OF PRECISION
* * *
1. You don’t literally need to use the back of an envelope of course; any scrap of paper will do. Which is just as well, because while I was writing this book, envelopes were removed from the list of essential items used in calculating the Retail Price Index, a sign that they are no longer a standard household item.
Back to text
2. For example, what interesting answer do you get if you multiply 3 × 7 × 11 × 13 × 37? Without a calculator, only somebody with strong mental calculation skills, an excess of curiosity, a lot of perseverance and a shortage of other things to do, would bother to find out. Even with a calculator in your pocket, you are still wondering if it’s worth the effort. (Go on, you know you want to.)