# 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.

So how should you start?

The first question asks for the fraction of the school children who liked tennis.

To work this out, you just need to take the following steps:

1. Put the number of degrees showing the tennis segment over 360 to create a fraction.
2. Simplify the fraction.

The number of degrees is 45, so the fraction is 45/360.

The first step to simplifying fractions is to see if the numerator goes into the denominator, which it does in this case: 45/45 = 1 and 360/45 = 8, so the fraction is 1/8 in its lowest terms.

(By the way, for a complete guide to simplifying fractions, just read Working with Fractions.)

The second question asks how many of the children preferred cricket.

To answer this, you should be able to learn a bit from the first question.

To work this out, you just need to take the following steps:

1. Put the number of degrees showing the cricket segment over 360 to create a fraction.
2. Multiply that fraction by the number of school children in the survey, which is 240.

As with the first question, you need to work out the fraction of the children in the survey you’re dealing with.

In this case, it’s 60/360 or 1/6.

To find out the number of children, you just have to multiply by 240, which is 1/6 x 240 = 40.

The final question asks you to estimate (or guess) how many of the children would say their favourite sport was football out of the whole school of 1200 pupils.

To work this out, you just need to take the following steps:

1. Work out the number of degrees taken up by the football segment of the pie chart.
2. Put the number of degrees over 360 to create a fraction.
3. Multiply that fraction by the number of children in the school, which is 1200.

To work out the number of degrees, it’s easier if you spot that the first half of the pie chart is composed of just football and tennis.

There are 180 degrees in total for that half, so taking away 45 degrees for the tennis-lovers gives you 135 degrees.

This works out at a fraction of 135/360 or 3/8.

Now, we only have data for the 240 children who’ve been surveyed, but that’s why we’re being asked to estimate the answer.

We have to assume that the other kids at school share the same preferences as the ones in the survey.

If we do that, all we need to do is multiply 3/8 by 1200 to get 3/8 x 1200 = 450.

And that’s it…!

# 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 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 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’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 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 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 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 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 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 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  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 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
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
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
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
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!
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 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
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 (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 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 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 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 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 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!
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 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 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 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 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
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
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
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 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 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 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 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 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 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 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
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 African field guide
Find an alphabetical list of the most common animals seen on safari in Africa, including mammals, reptiles and birds. more 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 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 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 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 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 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 Safari pub quiz
Challenge your friends and family on their wildlife knowledge with this fun quiz. more 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 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

# 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.

Adding and subtracting are usually the easiest sums, but not when it comes to fractions. If fractions have the same denominator (the number on the bottom), then you can simply add or subtract the second numerator from the first, eg 4/5 – 3/5 = 1/5. If not, it would be like adding apples and oranges.

They’re just not the same, so you first have to convert them into ‘pieces of fruit’ – or a common unit. The easiest way of doing that is by multiplying the denominators together. That guarantees that the new denominator is a multiple of both the others.

Once you’ve found the right denominator, you can multiply each numerator by the denominator from the other fraction (because whatever you do to the bottom of the fraction you have to do to the top), add or subtract them and then simplify and/or convert into a mixed number if necessary, eg 2/3 + 4/5 = (2 x 5 + 4 x 3) / (3 x 5) = (10 + 12) / 15 = 22/15 = 1 7/15.

1. Multiply the denominators together and write the answer down as the new denominator
2. Multiply the numerator of the first fraction by the denominator of the second and write the answer above the new denominator
3. Multiply the numerator of the second fraction by the denominator of the first and write the answer above the new denominator (after a plus or minus sign)
4. Add or subtract the numerators and write the answer over the new denominator
5. Simplify and/or turn into a mixed number if necessary

Note that you can often use a simpler method. If one of the denominators is a factor of the other, you can simply multiply the numerator and denominator of that fraction by 2, say, so that you get matching denominators, eg 1/5 + 7/10 = 2/10 + 7/10 = 9/10. This means fewer steps in the calculation and lower numbers, and that probably means less chance of getting it wrong.

### Sample Questions

1. 1/5 + 2/3
2. 3/8 + 11/12
3. 13/24 – 5/12
4. 7/8 – 3/4
5. 5/8 – 2/3

## Multiplication

This is the easiest thing to do with fractions. You simply have to multiply the numerators together, multiply the denominators together and then put one over the other, simplifying and/or converting into a mixed number if necessary, eg 2/3 x 4/5 = (2 x 4) / (3 x 5) = 8/15.

1. Multiply the numerators together
2. Multiply the denominators together
3. Put the result of Step 1 over the result of Step 2 in a fraction
4. Simplify and/or turn into a mixed number if necessary

1. 1/5 x 2/3
2. 7/12 x 3/8
3. 4/5 x 2/3
4. 4/9 x 3/4
5. 5/8 x 2/3

## Division

Dividing by a fraction must have seemed like a nightmare to early mathematicians, because nobody ever does it! That’s right. Nobody divides by a fraction, because it’s so much easier to multiply.

That’s because dividing by a fraction is the same as multiplying by the same fraction once it’s turned upside down, eg 2/3 ÷ 4/5 = 2/3 x 5/4 = (2 x 5) / (3 x 4) = 10/12 = 5/6. You can even cut out the middle step and simply multiply each numerator by the denominator from the other fraction, eg 2/3 ÷ 4/5 = (2 x 5) / (3 x 4) = 10/12 = 5/6.

1. Multiply the numerator of the first fraction by the denominator of the second
2. Multiply the numerator of the second fraction by the denominator of the first
3. Put the result of Step 1 over the result of Step 2 in a fraction
4. Simplify and/or turn into a mixed number if necessary

1. 1/5 ÷ 2/3
2. 2/7 ÷ 3/5
3. 4/7 ÷ 2/3
4. 7/8 ÷ 3/4
5. 5/6 ÷ 2/3

## Simplifying Fractions

One way of simplifying fractions is to divide by the lowest possible prime number over and over again, but that takes forever! It’s much simpler to divide by the Highest Common Factor (or HCF), which is either the numerator itself or half of it or a third of it etc:

1. If possible, divide both the numerator and the denominator by the numerator. If that works, you’ll end up with a ‘unit fraction’ (in other words, 1 over something) that can’t be simplified any more, eg 7/14 = 1/2 because 7 ÷ 7 = 1 and 14 ÷ 7 = 2.
2. If the numerator doesn’t go into the denominator, try the smallest fraction of the numerator (usually a half or a third) and then try to divide the denominator by the result, eg 24/36 = 2/3 because half of 24 is 12, and 36 ÷ 12 = 3.
3. If that doesn’t work, keep repeating Step 2 until you find the answer, eg 24/30 = 4/5 because a quarter of 24 is 6, and 30 ÷ 6 = 5 (and a half and a third of 24 don’t go into 30).

### Sample Questions

1. Simplify 14/28
2. Simplify 8/24
3. Simplify 30/50
4. Simplify 27/36
5. Simplify 45/72

## Turning Improper Fractions into Mixed Numbers

To turn an improper fraction into a mixed number, simply divide the numerator by the denominator to find the whole number and then put the remainder over the original denominator and simplify if necessary, eg 9/6 = 1 3/6 = 1 1/2.

1. Divide the numerator by the denominator
2. Write down the answer to Step 1 as a whole number
3. Put any remainder into a new fraction as the numerator, using the original denominator
4. Simplify the fraction if necessary

### Sample Questions

1. What is 22/7 as a mixed number?
2. What is 16/5 as a mixed number?
3. What is 8/3 as a mixed number?
4. What is 18/8 as a mixed number?
5. What is 13/6 as a mixed number?

## Turning Mixed Numbers into Improper Fractions

To turn a mixed number into an improper fraction, multiply the whole number by the denominator of the fraction and add the existing numerator to get the new numerator while keeping the same denominator, eg 2 2/5 = (10 + 2)/5 = 12/5.

1. Multiply the whole number by the denominator of the fraction
2. Add the answer to the existing numerator to get the new numerator
3. Write the answer over the original numerator
4. Simplify if necessary

### Sample Questions

1. What is 2 2/7 as an improper fraction?
2. What is 3 2/3 as an improper fraction?
3. What is 4 1/4 as an improper fraction?
4. What is 5 1/5 as an improper fraction?
5. What is 3 2/9 as an improper fraction?

There you go. Easy peasy lemon squeezy!

# 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.

They all do the same job of showing what share of something you have, and a common question involves converting from one to another, so here are a few tips…

## Fractions to Decimals

### Calculator

• Simply divide the numerator by the denominator, eg 3/4 = 3 ÷ 4 = 0.75.

### Non-calculator

You can always use the standard ‘bus stop’ method to divide the numerator by the denominator on paper (or in your head), but the numbers may be easy enough for you to use a shortcut.

• If the denominator is a power of 10 (eg 10 or 100), write the numerator down straight away as a decimal. You just have to make sure you end up with the digits in the right columns, eg a fraction involving hundredths needs to end in the second column after the decimal point, so 29/100 = 0.29.
• If the denominator ends in zero, you may be able to simplify the fraction into tenths first and then convert it into a decimal, eg 16/20 = 8/10 = 0.8.
• If you express the fraction in its lowest terms by simplifying it (ie dividing the numerator and denominator by the same numbers until you can’t go any further), you may  recognise a common fraction that you can easily convert, eg 36/45 = 4/5 = 0.8.

## Fractions to Percentages

### Calculator

• Simply divide the numerator by the denominator, multiply by 100 and add the ‘%’ sign, eg 3/4 = 3 ÷ 4 x 100 = 0.75 x 100 = 75%.

### Non-calculator

You can always convert the fraction into a decimal (see above) and then multiply by 100 and add the ‘%’ sign. Otherwise, try these short cuts in order.

• If the denominator is a factor of 100 (eg 10, 20, 25 or 50), multiply the numerator by whatever number will turn the denominator into 100 and add the ‘%’ sign, eg 18/25 = 18 x 4 = 72%.
• If the denominator is a multiple of 10 (eg 30, 40 or 70), divide the numerator by the first digit(s) of the denominator to turn the fraction into tenths, multiply the numerator by 10 and add the ‘%’ sign, eg 32/80 = 32 ÷ 8 x 10 = 4 x 10 = 40%.
• If you express the fraction in its lowest terms by simplifying it (ie dividing the numerator and denominator by the same numbers until you can’t go any further), you may  recognise a common fraction that you can easily convert from memory, eg 8/64 = 1/8 = 12.5%.

## Decimals to Fractions

Every decimal is really a fraction in disguise, so the method is the same whether you’re allowed a calculator or not.

### Calculator/non-calculator

• Check the final column of the decimal (eg tenths or hundredths) and place all the digits over the relevant power of 10 (eg 100 or 1000) before simplifying if necessary, eg 0.625 = 625/1000 = 5/8.

## Decimals to Percentages

Again, this is an easy one, so the method is the same whether you’re allowed a calculator or not.

### Calculator/non-calculator

• Multiply by 100 and add the ‘%’ sign, eg 0.375 x 100 = 37.5%.

## Percentages to Fractions

You can think of a percentage as simply a fraction over 100, so the method is easy enough whether you’re allowed a calculator or not.

### Calculator/non-calculator

• If the percentage is a whole number, remove the ‘%’ sign, place the percentage over 100 and simplify if necessary, eg 75% = 75/100 = 3/4. If not, turn the fraction into a whole number as you go by multiplying the numerator and denominator by whatever number you need to (usually 2, 3 or 4), eg 37.5% = (37.5 x 2) / (100 x 2) = 75/200 = 3/8.

## Percentages to Decimals

This is easy enough, so the method is the same whether you’re allowed a calculator or not.

### Calculator/non-calculator

• Remove the ‘%’ sign and divide by 100, eg 70% ÷ 100 = 0.7.