Maths on the Back of an Envelope: Clever Ways to (Roughly) Calculate Anything
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(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 pric
e 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
Di
stance 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.