Tag Archives: Maths shortcuts

Hints and Tips

Here are a few articles to show how to tackle common problems in English, Maths, French, Verbal and Non-verbal Reasoning and photography.

General

Hints and tips

 

 

 

 

 

How do I know if my child will get a place?
This is the question I get asked the most as a tutor. And even if parents don’t ask it directly, I know that it’s always lurking in the background somewhere…! more

English

Hard and soft g and c

 

 

 

 

Pronunciation
English is a funny old language. It’s such a mishmash of imported words and complicated constructions that it was once described as having French vocabulary and German grammar! Unfortunately, that means the spelling and pronunciation of words are often different. Two of the letters that cause problems are c and g. more

 

 

 

 

 

 

Why I hate the Press!
I know why they do it (most of the time), but it’s still incredibly annoying and confusing. I’m talking about grammatical mistakes in the papers. more

 

 

 

 

 

Americanisms
In the words of Winston Churchill (or George Bernard Shaw or James Whistler or Oscar Wilde), Britain and America are “two nations divided by a single language”. Quite a few of my pupils live outside the United Kingdom and/or go to foreign schools but are applying to English schools at 11+ or 13+ level. One of the problems they face is the use of Americanisms. more

 

 

 

 

 

Colons and semicolons
Using colons and semicolons is often an easy way to get a tick in your homework, but it still involves taking a bit of a risk. If you get it right, you get the tick, but if you get it wrong, you’ll get a cross. This article will explain how to use both colons and semicolons so that you can be confident of getting far more ticks than crosses! more

 

 

 

 

 

 

Explaining humour
The ‘W’ words are useful if you’re trying to understand or summarise a story, but who, whom, who’s and whose tend to cause problems. Here’s a quick guide to what they all mean and how they can be used. more

Who-or-whom

Who or whom, who’s or whose?
The ‘W’ words are useful if you’re trying to understand or summarise a story, but whowhomwho’s and whose tend to cause problems. Here’s a quick guide to what they all mean and how they can be used. more

Could vs might

Could or might?
Could and might mean different things, but a lot of people use them both to mean the same thing. Here’s a quick guide to avoid any confusion. more

Homophones

Homophones
Homophones are words that sound the same even though they’re spelt differently and mean different things. Getting them right can be tricky, but it’s worth it in the end. more

Creating off-the-shelf characters
Common entrance exams have a time limit. If they didn’t, they’d be a lot easier! If you want to save time and improve your story, one thing you can do is to prepare three ‘off-the-shelf’ characters that you can choose from. more

Books
Children’s reading list
I’m often asked by parents what books their children should be reading. Here’s a list of my favourite books when I was a boy. Maybe a few of them might be worth ordering online…! more

John McEnroe
Describing feelings
In many 11+ and 13+ exams, you have to talk about feelings. Yes, I know that’s hard for most boys that age, but I thought it might help if I wrote down a list of adjectives that describe our emotions. Here we go… more

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How to write a letter
Writing a letter is not as easy as it might seem – especially if you have to do it during a Common Entrance exam! In this post, I’d like to explain the typical format of formal and casual letters and the decisions on wording that you’ll have to make… more

Grand Central
Descriptive writing
Exams at 11+ and 13+ level always let you tell a story in the writing section, but they sometimes provide a picture and simply ask you to describe it or to ‘write about it in any way you like’. Writing a description is obviously different from writing a story, so it’s worthwhile pointing out the differences and the similarities… more

SVO
What is a full sentence?
Teachers often tell pupils to use a ‘full sentence’ in their answers, but what is a full sentence? more

Apostrophe
It’s all about the apostrophe
The apostrophe is tricky. It means different things at different times. This article is meant to clear up any confusion and help you use apostrophes, which might mean you get straight As in your exams – or should that be A’s?! more

Keep calm and check your spelling
Spelling rules
The problem with the English is that we’ve invaded (and been invaded by) so many countries that our language has ended up with a mish-mash of spelling rules… more

 

 

 

 

 

 

 

Parts of speech
English exams often ask questions about the ‘parts of speech’. This is just a fancy term for all the different kinds of words, but they’re worth knowing just in case. Just watch out for words such as ‘jump’, which can be more than one part of speech! more

Letter N
Capital!
The three main things to check after writing anything are spelling, punctuation and capital letters, so when do you use capitals? more

speech marks
Speech marks
Speech marks, inverted commas, quotation marks, quote marks, quotes, 66 and 99 – does any other punctuation mark have so many names or cause so much confusion…?! more

Essay writing
Essay writing
There comes a point in everyone’s life when you have to undergo the ritual that marks the first, fateful step on the road to becoming an adult. It’s called ‘writing an essay’… more

Commas
Commas
If you had the chance to take a contract out on one punctuation mark, most people would probably choose the comma. Unfortunately, that’s not possible, although modern journalists are doing their best to make it into an optional extra… more

Poetic devices
Poetic devices
It’s important to be able to recognise and analyse poetic devices when studying literature at any level. Dylan Thomas is my favourite poet, and he uses so many that I decided to take most of my examples from his writings… more

Story mountains
Story mountains
Everyone needs a route map, whether it’s Hillary and Tenzing climbing Mount Everest or an English candidate writing a story. One of the ways of planning a story is to create a story mountain, with each stage of the tale labelled on the diagram… more

Remember the iceberg!
Remember the iceberg!
To pass Common Entrance, you have to be a scuba diver. Only a small part of any iceberg is visible above the waves, and only a small part of any answer to a question is visible in the text. To discover the rest, you have to ‘dive in’ deeper and deeper… more

Maths

Number triangle

 

 

 

 

 

 

 

Number Triangles

Number triangles are a useful way of working out how to rearrange a multiplication or division sum. This is important if you have to ‘fill in the gaps’, for example. more

Problem Questions

 

 

 

 

 

 

Problem Questions
‘Problem questions’ are often the most difficult in 11+ and 13+ Maths papers.
There are several different kinds, but they all have one thing in common: they all ‘hide’ the sums that you have to do.
That means the first thing you have to do is work out the actual calculations you’re being asked for. more

Value of pi

 

 

 

 

 

Rounding
Rounding is just a convenient way of keeping numbers simple. Nobody wants to have to remember all the decimals in 𝝅 (which is 3.1415926535897932384…), so people usually round it to 3.14 (or 22/7).

There are three ways of rounding numbers:

  • using a power of 10
  • using decimal places
  • using significant figures. more

 

Working out values from a pie chart

Working out values from a pie chart
This is a typical question from a Dulwich College 11+ Maths paper that asks you to work out various quantities from a pie chart. To answer questions like this, you have to be comfortable working with fractions and know that there are 360 degrees in a circle. more

Reflecting shapes in a mirror line

Reflecting shapes in a mirror line
This is a typical question from a Dulwich College 11+ Maths paper, and it asks you to draw a reflection of the triangle in the mirror line shown on the chart. more

SOHCAHTOA
SOHCAHTOA
SOHCAHTOA (pronounced ‘soccer-toe-uh’) is a useful ‘mnemonic’ to remember the definitions of sines, cosines and tangents. Amazingly, I was never taught this at school, so I just had to look up all the funny numbers in a big book of tables without understanding what they meant! more

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Long multiplication
You can use short multiplication if you’re multiplying one number by another that’s in your times tables (up to 12). However, if you want to multiply by a higher number, you need to use long multiplication. more

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How to add, subtract, multiply and divide
The most important things you need to do in Maths are to add, subtract, divide and multiply. If you’re doing an entrance exam, and there’s more than one mark for a question, it generally means that you have to show your working. more

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Long division
Long division is on the syllabus for both 11+ and 13+ exams, so it’s important to know when and how to do it… more

maths trick
Maths trick
Here’s a Maths trick a friend of mine saw on QI. Who knows? It might make addition and subtraction just a little bit more fun! more

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Simultaneous equations
Why do we have simultaneous equations? Well, there are two ways of looking at it… more

Prime factors
Prime factors
Prime factors have nothing to do with Optimus Prime – sadly – but they often crop up in Maths tests and can be used to find the Lowest Common Multiple or Highest Common Factor of two numbers… more

negative-numbers
Negative numbers
Working with negative numbers can be confusing, but a few simple rules can help you add, subtract, multiply and divide successfully… more

maths
Useful terms in Maths
Maths is complicated, but a good first step on the road to understanding it is to get to know the most useful terms. There are lists in the front of the Bond books, but here’s my own contribution. I hope it helps! more

algebra
Algebra
Algebra is supposed to make life easier. By learning a formula or an equation, you can solve any similar type of problem whatever the numbers involved. However, an awful lot of students find it difficult, because letters just don’t seem to ‘mean’ as much as numbers. Here, we’ll try to make life a bit easier… more

Divisibility rules OK
Divisibility rules OK!
Times tables can be tricky, and there’s no substitute for learning them by heart. However, the divisibility rules can at least tell you whether an answer is definitely wrong. I’m a great believer in ‘sanity checking’ your work. Just ask yourself, “Is this crazy?” If it is, you’ll have to do the question again! more

Back-to-school-blackboard-chalk
Tips for the QTS numeracy test
The QTS numeracy and literacy tests are not very popular, but trainee teachers still have to pass them before they can start teaching in the state sector, so I thought I’d try and help out. There is always more than one way of doing a Maths question, but I hope I’ll demonstrate a few useful short cuts and describe when and how they should be used… more

Here be ratios
Ratios
Hundreds of years ago, it was traditional to put dragons on maps in places that were unknown, dangerous or poorly mapped. Ratios are one of those places… more

Fractions, decimals and percentages
Working with fractions
People don’t like fractions. I don’t know why. They’re difficult to begin with, I know, but a few simple rules will help you add, subtract, multiply and divide… more

Number sequences
Number sequences
Number sequences appear in Nature all over the place, from sunflowers to conch shells. They can also crop up either in Maths or Verbal Reasoning, and both are essential parts of 11+ and other school examinations… more

Fractions, decimals and percentages
Fractions, decimals and percentages
Pizzas are very useful, mathematically speaking. However much we hate fractions, we all know what half a pizza looks like, and that’s the point. Numbers don’t have any intrinsic meaning, and we can’t picture them unless they relate to something in the real world, so pizzas are just a useful way of illustrating fractions, decimals and percentages… more

Useful formulas
Useful formulas
What is a problem? A problem = a fact + a judgment. That is a simple formula that tells us something about the way the world works. Maths is full of formulas, and that can intimidate some people if they don’t understand them or can’t remember the right one to use… more

Short cuts
Short cuts
There is always more than one way of solving a Maths problem. That can be confusing, but it can also be an opportunity – if only you can find the right trade-off between speed and accuracy… more

French

French verbs
French regular verbs – present subjunctive tense
The subjunctive in French is generally used in the present tense after expressions such as ‘il faut que’ and certain verbs that also take the word ‘que’ after them. These are generally the ones that express feelings or doubts (eg craindre, vouloir), especially when two parts of a sentence have different subjects, eg ‘I want her to be happy’ becomes ‘Je veux qu’elle soit contente’. Verbs ending in -er or -re have one set of endings, but  -ir verbs have another… more

French verbs
Preceding Direct Objects in French
Forming the perfect (or pluperfect) tense in French is sometimes made harder than necessary by what’s called a Preceding Direct Object (or PDO). The object of a sentence is whatever ‘suffers the action of the verb’, eg the nail in ‘he hit the nail on the head’… more

French verbs
French regular verbs – conditional tense
The conditional tense in French is used to show that someone ‘would do’ or ‘would be doing’ something. All verbs end in -er, -re or -ir, and the endings are different (as shown here in red)… more

French verbs
French regular verbs – future tense
There is only one future tense in French, and it’s used to show that someone ‘will do’ or ‘will be doing’ something. Verbs end in -er, -re or -ir, but the endings are the same… more

French verbs
French regular verbs – past tense
Here are the basic forms of French regular verbs in the past tense, which include the perfect (or passé composé), pluperfect, imperfect and past historic (or passé simple). All verbs end in -er, -re or -ir, and there are different endings for each that are shown here in red… more

French verbs
Common French verbs – present tense
Language changes over time because people are lazy. They’d rather say something that’s easy than something that’s correct. That means the most common words change the most, and the verbs become ‘irregular’. In French, the ten most common verbs are ‘être’, ‘avoir’, ‘pouvoir’, ‘faire’, ‘mettre’, ‘dire’, ‘devoir’, ‘prendre’, ‘donner’ and ‘aller’, and they’re all irregular apart from ‘donner’… more

French verbs
French regular verbs – present tense
Nobody likes French verbs – not even the French! – but I thought I’d start by listing the most basic forms of the regular verbs in the present tense. All French verbs end in -er, -re or -ir, and there are different endings for each that are shown here in red… more

Learning the right words
Learning the right words
One of the frustrations about learning French is that you’re not given the words you really need to know. I studied French up to A-level, but I was sometimes at a complete loss when I went out with my French girlfriend and a few of her friends in Lyon. I was feeling suitably smug about following the whole conversation in French…until everyone started talking about chestnuts! more

Non-verbal Reasoning

Non-verbal Reasoning
Non-verbal reasoning tests are commonly found in Common Entrance exams at 11+ and 13+ level, and they’re designed to test pupils’ logical reasoning skills using series of shapes or patterns. It’s been said that they were intended to be ‘tutor-proof’, but, of course, every kind of test can be made easier through proper preparation and coaching. more

Photography

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African field guide
Find an alphabetical list of the most common animals seen on safari in Africa, including mammals, reptiles and birds. more

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Basics of photography
Learn all about the basic aspects of photography, including types of camera, types of lenses, the Exposure Triangle (shutter speed, aperture and ISO), focus and other settings. more

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Game drives
Read all about the best gear, equipment to take with you on safari, learn the rules of composition and find out the best workflow for editing your wildlife images. more

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How to stand out from the herd
Read this quick guide to improve your wildlife shots by setting up something a little bit different, from slow pans to sunny silhouettes. more

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Introduction to Lightroom
Learn how to import, edit and organise your images in Lightroom, including the main features available in the Library and Develop modules and a summary of keyboard shortcuts. more

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Making money from photography
Find out how to start making money from your photography with this quick and easy guide to entering competitions, putting on exhibitions, selling through stock (and microstock) agencies and more. more

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Rules of composition
Find out the rules of composition to help you get the most out of your photography, including the Rule of Thirds, framing, point of view, symmetry and a whole lot more. more

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Safari pub quiz
Challenge your friends and family on their wildlife knowledge with this fun quiz. more

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Wildlife photography
Learn how to take great wildlife shots by preparing properly, taking the right equipment and getting to know the rules of composition. more

Verbal Reasoning

VRTypeO

Verbal Reasoning
Verbal Reasoning (VR) tests were invented to test pupils’ logic and language skills – although they do sometimes includes questions about numbers. In order to do well in a VR test, the most important thing is to be systematic, to have a plan for what to do if the question is hard. Here is a guide to the different kinds of problems and the best ways to approach them. more

Shortcuts

Short cuts

It’s just through here…

There is always more than one way of solving a Maths problem. That can be confusing, but it can also be an opportunity – if only you can find the right trade-off between speed and accuracy.

I’ve taught a lot of QTS numeracy candidates recently, and the Maths itself isn’t particularly difficult, particularly in the mental arithmetic section.

The trick is to be familiar with all the possible short cuts and capable of using the right one at the right time. It may mean having to do more sums, but it will be much simpler and quicker in the long run. You don’t have to use all of these all the time, but it is useful to know what they are just in case you need them.

  • Multiplying and dividing by 5
    The most useful short cut I’ve come across is very simple. To multiply by 5, try multiplying by 10 and then dividing by 2 (or vice versa), eg
    13 x 5
    = 13 x 10 ÷ 2
    = 130 ÷ 2
    = 65
    You have to do two sums rather than one, but the point is that you should be able to save time and improve the chances of getting the answer right by doing both in your head rather than having to work out a more difficult sum on paper.
    You can do divide by 5 in a similar way by multiplying by 2 and dividing by 10 (or vice versa), eg
    65 ÷ 5
    = 65 x 2 ÷ 10
    = 130 ÷ 10
    = 13
    You can do a similar trick with 50, 500 etc simply by multiplying or dividing by a higher power of ten.
  • Chunking
    If you have to multiply by a two-digit number outside your times tables, chunking is an easy way to do the sum in your head. Instead of writing it down on paper and using long multiplication (which would take a long time and is easy to get wrong!), try multiplying by the tens and the units separately and adding up the results, eg 16 x 15 = 10 x 15 + 6 x 15 = 150 + 90 = 240. The numbers might still be too tricky to do it comfortably, but it’s often worth a try.
  • Rounding
    To avoid sums with ‘tricky’ numbers, try rounding them up to the nearest ‘easy’ figure and adjusting at the end. This is particularly useful when working out start and end times, eg
    ‘The morning session in a school began at 09:25. There were three lessons of 50
    minutes each and one break of 20 minutes. At what time did the morning session end? Give your answer using the 24-hour clock.’
    If you assume the lessons last an hour, you can add three hours to 09:25 to get 12:25. You would normally then knock off 3 x 10 = 30 minutes, but the 20-minute break means you only need to subtract 10 minutes, which means the session ended at 12:15.
  • Money problems
    There is often a ‘real world’ money problem in the QTS numeracy test. That usually means multiplying three numbers together. The first thing to say is that it doesn’t matter in which order you do it – 1 x 2 x 3 is just the same as 3 x 2 x 1. The next thing to bear in mind is that you will usually have to convert from pence to pounds. You could do this at the end by simply dividing the answer by 100, but a better way is to divide one of the numbers by 100 (or two of the numbers by 10) at the beginning or turn multiplication by a fraction of a pound into a division sum, eg
    ‘All 30 pupils in a class took part in a sponsored spell to raise money for charity. The pupils were expected to get an average of 18 spellings correct each. The average amount of sponsorship was 20 pence for each correct spelling. How many pounds would the class expect to raise for charity?’
    The basic sum is 30 x 18 x 20p, and there are a couple of ways you could do this:
    1) Knock off the zeroes in two of the numbers, change the order of the numbers to make it easier and double and halve the last pair to give yourself a sum in your times tables, ie
    30 x 18 x 20p
    = 3 x 18 x 2
    = 3 x 2 x 18
    = 6 x 18
    = 12 x 9
    = £108
    2) Convert pence into pounds, turn it into a fraction, change the order of the numbers, divide by the denominator and, again, double and halve the last pair to give yourself a sum in your times tables, ie
    30 x 18 x 20p
    = 30 x 18 x £0.20
    = 30 x 18 x ⅕
    = 30 x 18 ÷ 5
    = 30 ÷ 5 x 18
    = 6 x 18
    = 12 x 9
    = £108
  • Percentages
    Many students get intimidated by percentages, fractions and decimals, but they are all just different ways of showing what share you have of something. You will often by asked to add or subtract a certain percentage. The percentage will usually end in zero (eg 20%, 30% or 40%), so the easiest way is probably to find 10% first. That just means dividing by 10, which means moving the decimal point one place to the left or, if you can, knocking off a zero. Once you know what 10% is, you can simply multiply by 2, 3 or 4 etc and add or subtract that number to find the answer, eg
    ‘As part of the numeracy work in a lesson, pupils were asked to stretch a spring to extend its length by 40 per cent. The original length of the spring was 45 centimetres. What should be the length of the extended spring? Give your answer in centimetres.’
    You need to find 40% of 45cm, so you can start by finding 10%, which is 45 ÷ 10 or 4.5cm. You can then multiply it by 4 to find 40%, which is best done by doubling twice, ie 4.5 x 2 x 2 = 9 x 2 = 18. Finally, you just add 18cm to the original length of the spring to find the answer, which is 45 + 18 = 63cm.
  • Common fractions
    An awful lot of questions involve converting between fractions, percentages and decimals. There is a proper technique for doing any of those, but it’s very useful if you learn the most common fractions and their decimal and percentage equivalents by heart, eg
    ½ = 0.5 = 50%
    ¼ = 0.25 = 25%
    ¾ = 0.75 = 75%
    ⅕ = 0.2 = 20%
    ⅖ = 0.4 = 40%
    ⅗ = 0.6 = 60%
    ⅘ = 0.8 = 80%
    ⅛ = 0.125 = 12.5%
    ⅜ = 0.375 = 37.5%
    ⅝ = 0.625 = 62.5%
    ⅞ = 0.875 = 87.5%
  • Times tables
    There are far more multiplication questions in the QTS numeracy test than any other kind, so it’s very important to know your times tables inside out. Some pupils are taught to memorise only the results, eg 4, 8, 12… etc. This is catastrophic! If you have to go through the whole table to find the answer, counting off the number of fours on your fingers, you can’t save yourself any time at all. The proper way is to learn the whole sum, eg 1 x 4 is 4, 2 x 4 is 8, etc (or 1 4 is 4, 2 4s are 8, etc). That way, the answer to any question in your times tables will pop into your head as soon as you’ve heard it. One good way of learning your tables is to time yourself using the stopwatch function on your iPhone. If you press ‘Lap’ after you’ve recited each table, you can write down your times and work out which tables you need to practise. Once you’re confident, you can make certain sums fit into your times tables by doubling one number and halving the other, eg
    3 x 24
    = (3 x 2) x (24 ÷ 2)
    = 6 x 12
    = 72
    Alternatively, you can halve just one of the numbers and double the result, eg
    24 x 9
    = 12 x 9 x 2
    = 108 x 2
    = 216
  • Multiplying by 4
    If you have to multiply by 4 and the number is not in your times tables, a simple way to do it is to double it twice, eg
    26 x 4
    = 26 x 2 x 2
    = 52 x 2
    = 104
  • Multiplying by a multiple of 10
    If you have to multiply by a multiple of 10 such as 20 or 30, try knocking the zero off and adding it in again afterwards. That way, you don’t have to do any long multiplication and, with any luck, the sum will be in your times tables, eg
    12 x 30
    = 12 x 3 x 10
    = 36 x 10
    = 360
  • Multiplying decimals
    This can be a bit confusing, so the best way of doing it is probably to ignore any decimal points, multiply the numbers together and then add back the decimal point to the answer so that you end up with the same number of decimal places as you had in the beginning, eg
    0.5 x 0.5
    = 5 x 5 ÷ 100
    = 25 ÷ 100
    = 0.25
  • Using the online calculator
    The second section of the QTS numeracy test consists of on-screen questions that can be answered using an online calculator. This obviously makes working out the answer a lot easier, and short cuts are therefore less useful. However, just because the calculator’s there doesn’t mean you have to use it, particularly for multiple-choice questions. If you have to add up a column of cash values, for example, and compare it with a number of options, you could simply tot up the number of pence and pick the option with the right amount. Alternatively, the level of accuracy needed in the answer may give you a helping hand if it rules out all but one of the possible answers, eg 6 ÷ 21 to one decimal place is always going to be 0.3. Why? Well, it’s a bit less than 7 ÷ 21, which would be a third or 0.3 recurring. An answer of 0.4 would be more than that, and 0.2 would be a fifth, which is far too small, so it must be 0.3.
  • Don’t do more than you have to!
    There are several types of question that could tempt you into doing more work than you need to do. If you’re trying to work out how many tables you need at a wedding reception for a given number of guests, the answer is always going to need rounding up to the next whole number, so you don’t need to spend any time working out the exact answer to one or two decimal places. Equally, some numbers are so close to being an ‘easy’ number that you don’t need to add or subtract anything after rounding up or down to make the basic sum easier, eg
    ‘For a science experiment a teacher needed 95 cubic centimetres of vinegar for each pupil. There were 20 pupils in the class. Vinegar comes in 1000 cubic centimetre
    bottles. How many bottles of vinegar were needed?
    If you round 95cc to 100cc, the answer is 20 x 100 ÷ 1000 or 2 bottles, and the remainder consisting of 20 lots of 5cc of vinegar can safely be ignored.