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Explaining humour

Explaining humourOne of the things that children taking Common Entrance exams at either 11+ or 13+ find most difficult to explain is humour. Here’s a quick guide to various different types with explanations, examples and a short quiz at the end.

Slapstick comedy or farce

This is a type of physical comedy that relies on the fact that we find it funny when other people hurt themselves. It’s called ‘Schadenfreude’ in German, and it really shouldn’t be funny…but it is!

Example: A man slips on a banana skin and falls over.

Deadpan or dry humour

This is any joke that’s told with a very matter-of-fact tone.

Example: “It can hardly be a coincidence that no language on earth has ever produced the expression ‘As pretty as an airport’.”
The Long Dark Tea-time of the Soul, by Douglas Adams


This means putting oneself down in a self-mocking way.

Example: “If a book about failures doesn’t sell, is it a success?”
Jerry Seinfeld

Toilet and bodily humour

What we do in the toilet or in the bedroom has given rise to a LOT of jokes over the years…

Example: “It’s just a penis, right? Probably no worse for you than smoking.”
When You Are Engulfed in Flames, by David Sedaris

Puns, wit and wordplay

These are jokes based on double meanings or a play on words.

Example: “If not actually disgruntled, he was far from being gruntled.”
The Code of the Woosters, by P.G. Wodehouse


An epigram is just a saying, and some sayings can be very funny – whether deliberately or not!

Example: “Always go to other people’s funerals, otherwise they won’t come to yours.”
Yogi Berra

Dark humour

Dark humour is usually about death or the gloomier aspects of life.

Example: I come from Des Moines. Somebody had to.”
The Lost Continent: Travels in Small-Town America, by Bill Bryson

Sarcasm and irony

Sarcasm is saying exactly the opposite of what you mean, but irony is much richer and more popular because the meaning for the reader can be anything from the literal truth of the statement to its exact opposite. It’s up to you…

Example: “It is a truth universally acknowledged that a single man in possession of a good fortune must be in want of a wife.”
Pride and Prejudice, by Jane Austen


Finding a rude double meaning in a word or phrase is called innuendo.

Example: “Headline?” he asked.
“‘Swing Set Needs Home,'” I said.
“‘Desperately Lonely Swing Set Needs Loving Home,'” he said.
“‘Lonely, Vaguely Pedophilic Swing Set Seeks the Butts of Children,'” I said.”
The Fault in Our Stars, by John Green


This expression just means the writer or speaker is being insincere in an ironic and/or mocking way.

Example: “In the beginning, the Universe was created. This has made a lot of people very angry and been widely regarded as a bad move.”
The Hitchhiker’s Guide to the Galaxy, by Douglas Adams

Exaggeration and hyperbole

Exaggeration can lead to a powerful punchline in a joke because it relies on shocking the reader with something unexpected.

Example: “In our family, there was no clear line between religion and fly fishing.”
A River Runs Through It, by Norman Maclean

Parody and mockery

Pretending to write in a certain style or copying the format of a particular writer or type of text can be done humorously – although the implied criticism may be affectionate.

Example: “It is a truth universally acknowledged that a zombie in possession of brains must be in want of more brains.”
Pride and Prejudice and Zombies, by Seth Grahame-Smith and Jane Austen


This is making fun of something usually in religion, politics or current affairs.

Example: “They say the world is flat and supported on the back of four elephants who themselves stand on the back of a giant turtle.”
The Fifth Elephant, by Terry Pratchett

The surreal

‘Surreal’ just means absurd, nightmarish or like a fantasy.

Example: “As Gregor Samsa awoke one morning from uneasy dreams he found himself transformed in his bed into a gigantic insect.”
The Metamorphosis, by Franz Kafka

Character humour

Like a lot of sit-coms this form of humour relies on the personality of the characters. Things are funny because they are so typical of a certain type of person – often a stereotype.

Example: “As a boy, I wanted to be a train.”
Machine Man, by Max Barry


A lot of stand-up comedy is based on observational humour, which means simply picking up on the typical habits of people in the world around us. We laugh because we recognise the behaviour and often the reason for it.

Example: “It’s a funny thing about mothers and fathers. Even when their own child is the most disgusting little blister you could ever imagine, they still think that he or she is wonderful.”
Matilda, by Roald Dahl


The shock value of an insult lends itself to humour.

Example: Two whales walk into a bar. The first whale says to the other, “WOOOOOO. WEEEEEEEEOOOOO. WEEEEEEEEEEEEOOOOOOOOO.” The second whale says, “Shut up Steve, you’re drunk.”

Awkward situations

If a situation is particularly cringeworthy or awkward, then it will often generate nervous laughter.

Example: “I don’t know how other men feel about their wives walking out on them, but I helped mine pack.”
Breaking Up, by Bill Manville

Blue or off-colour jokes

Using rude words or swear words has the shock value that can generate humour.

Example: “If this typewriter can’t do it, then f*** it, it can’t be done.”
Still Life With Woodpecker, by Tom Robbins

Sample questions

How would you explain the humour in these lines?

  1. “An unhappy alternative is before you, Elizabeth. From this day, you must be a stranger to one of your parents. your mother will never see you again if you do not marry Mr Collins, and I will never see you again if you do.”
    Pride & Prejudice, by Jane Austen
  2. “There’s a door,” he whispered.
    “Where does it go?”
    “It stays where it is, I think,” said Rincewind.
    Eric, by Terry Pratchett
  3. “It’s not because I want to make out with her.”
    “Hold on.”
    He grabbed a pencil and scrawled excitedly at the paper as if he’d just made a mathematical breakthrough and then looked back up at me.
    “I just did some calculations, and I’ve been able to determine that you’re full of s**t.”
    Looking for Alaska, by John Green
  4. “I came from a real tough neighborhood. Once a guy pulled a knife on me. I knew he wasn’t a professional: the knife had butter on it.”
    Rodney Dangerfield
  5. “A word to the wise ain’t necessary. It’s the stupid ones who need advice.”
    Bill Cosby
  6. “To win back my youth, Gerald, there is nothing I wouldn’t do – except take exercise, get up early or be a useful member of the community.”
    A Woman of No Importance, by Oscar Wilde
  7. “Some men are born mediocre, some men achieve mediocrity, and some men have mediocrity thrust upon them. With Major Major, it had been all three. Even among men lacking all distinction, he inevitably stood out as a man lacking more distinction than all the rest, and people were always impressed by how unimpressive he was.”
    Catch-22, by Joseph Heller
  8. “Build a man a fire, and he’ll be warm for a day. Set a man on fire, and he’ll be warm for the rest of his life.”
    Jingo, by Terry Pratchett
  9. “There are moments, Jeeves, when one asks oneself, ‘Do trousers matter?'”
    “The mood will pass, sir.”
    The Code of the Woosters, by PG Wodehouse
  10. “There was a boy called Eustace Clarence Scrubb, and he almost deserved it.”
    The Voyage of the Dawn Treader, by CS Lewis
  11. “I write this sitting in the kitchen sink.”
    I Capture the Castle, by Dodie Smith
  12. “You can lead a horticulture, but you can’t make her think.”
    Dorothy Parker
  13. “For a moment, nothing happened. Then, after a second or so, nothing continued to happen.”
    The Hitchhiker’s Guide to the Galaxy, by Douglas Adams
  14. “For the better part of my childhood, my professional aspirations were simple – I wanted to be an intergalactic princess.”
    Seven Up, by Janet Evanovich
  15. “It wasn’t until I had become engaged to Miss Piano that I began avoiding her.”
    Into Your Tent I’ll Creep, by Peter De Vries
  16. “To lose one parent, Mr. Worthing, may be regarded as a misfortune; to lose both looks like carelessness.”
    The Importance of Being Earnest, by Oscar Wilde

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.

  • Write down the numbers one on top of the other with the smaller number on the bottom and a times sign on the left (just as you would normally), then draw three lines underneath to hold three rows of numbers.
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  • Multiply the top number by the last digit of the bottom number as you would normally.
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  • Write a zero at the end of the next answer line (to show that you’re multiplying by tens now rather than units).
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  • Multiply the top number by the next digit of the bottom number, starting to the left of the zero you’ve just added.
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  • Add the two answer lines together to get the final answer.
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  • Some people write the tens they’ve carried right at the top of the sum, but that can get very confusing with three lines of answers!
  • Don’t forget to add the zero to the second line of your answer. If it helps, you can try writing it down as soon as you set out the sum (and before you’ve even worked anything out).
  • At 11+ level, long multiplication will generally be a three-digit number multiplied by a two-digit number, but the method will work for any two numbers, so don’t worry. If you have to multiply two three-digit numbers, say, you’ll just have to add another line to your answer.

Sample questions:

Have a go at these questions. Make sure you show your working – just as you’d have to do in an exam.

  1. 216 x 43
  2. 17 x 423
  3. 23 x 648
  4. 782 x 28
  5. 127 x 92

Who or whom, who’s or whose?

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.

Who v whom

Who and whom are both relative pronouns, which mean they relate to the person you’ve just been talking about. Note that they don’t relate to animals or things, just people. The difference is just one letter, but it signals that one of them stands for the subject (in the nominative case if you’ve ever done Latin) while the other stands for the object (in the accusative).

  • The subject of a sentence is the noun or pronoun that controls the verb, in other words the person or thing that’s ‘doing the doing’.
  • The object of a sentence is the noun or pronoun that is suffering the action the verb, in other words the person or thing that’s having something done to it.

For example, in the following sentence, ‘the girl’ is the subject, and ‘the boy’ is the object:

The girl tapped the boy on the shoulder.

We could also use pronouns, in which case ‘she’ is the subject, and ‘him’ is the object.

She tapped him on the shoulder.

Note that we use ‘him’ rather than ‘he’ in this case. That tells us that the boy is the object and not the subject. It’s the same with ‘who’ and ‘whom’. In fact, it’s the same letter – the letter ‘m’ – that tells us that ‘him’ and ‘whom’ are both the objects of the sentence, and that might be a good way to remember the difference.

For example, in the following sentence, ‘the girl’ is still the subject, so we use ‘who’:

They saw the girl who had tapped the boy on the shoulder.

In the next sentence, the boy is still the object, so we use ‘whom’:

They saw the boy whom the girl had tapped on the shoulder.

Note that neither who nor whom needs a comma before it in these cases. That’s because we are defining which people we’re talking about. It’s a bit like ‘which’ and ‘that’: ‘which’ describes things and needs a comma, but ‘that’ defines things and doesn’t. If we already know who people are and simply want to describe them, then we do use a comma.

They saw Patricia Smith, who had tapped the boy on the shoulder.

They saw Paul Jones, whom the girl had tapped on the shoulder.

In these cases, we know who the children are – Patricia and Paul – so all we’re doing is describing something that has happened. There is only one Patricia Smith and one Paul Jones, so we don’t need to define them. That means we need to use a comma in both cases.

I hope that all makes sense. Here are a few practice questions. Just decide in each case whether you should use ‘who’ or ‘whom’.

  1. They talked to Jim, who/whom lived in Stoke.
  2. He played football with the boy who/whom had red hair.
  3. She was friends with the girl who/whom played volleyball.
  4. Who/whom do you think will win the egg and spoon race?
  5. Who/whom did they put in prison?

Who’s v whose

The words ‘who’s’ and ‘whose’ are homophones, which is another way of saying they sound the same but mean completely different things. ‘Who’s’ is short for ‘who is’ or ‘who has’ while ‘whose’ is a possessive pronoun that means ‘of whom the’ or ‘of which the’. For example, take these two sentences:

  • Who’s going to the cinema tonight?
  • He was a big man whose hands were larger than dinner plates.

The first means ‘Who is going to the cinema tonight?’ whereas the second means ‘He was a big man of whom the hands were larger than dinner plates’. The only reason we don’t say those things is that they’re a bit of a mouthful, so it’s easier to use ‘who’s’ or ‘whose’.

I hope that’s clear now. Here are a few practice questions. Just decide in each case whether you should use ‘who’s’ or ‘whose’.

  1. Who’s/whose in charge of the tennis rackets?
  2. Who’s/whose bag is this?
  3. He speaks to the woman who’s/whose behind the counter.
  4. She likes him to know who’s/whose boss.
  5. Who’s/whose been eating all the crisps?


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.

The reason why homophones are important is not just to do with the general need to spell correctly. Many people think getting them wrong is a ‘worse’ mistake than simply mis-spelling a word because it means that you don’t really know what you’re doing. Anyone can make a spelling mistake, but using completely the wrong word somehow seems a lot worse. That may not sound fair, but that’s just how a lot of people think, so it’s worth learning the common homophones so you don’t get caught out.

Here’s a list of the main ones:

a lot/allotmuch/distribute
ad/addnotice/put together
ads/adds/adzenotices/puts together/type of axe
ail/aletrouble (verb)/beer
air/heiratmospheric gas/inheritor
aisle/isle/I’llcorridor/island/I will
all ready/alreadyeverything set/by a certain time
all together/altogethereveryone in the same place/absolutely
all/awltotally/piercing tool
allowed/aloudpermitted/out loud
alter/altarchange/church table
ant/auntinsect/parent’s sister or brother’s wife
arc/arkpart of circle/Biblical boat
aural/oralto do with hearing/to do with mouths
away/aweighoff/up (eg anchors aweigh)
ay/aye/eye/Iyes/yes/organ of sight/1st person
bail/balescoop water out/jump out
bait/batefood on hook/hold (eg bated breath)
bare/bearnaked/tolerate or grizzly
base/bassfoundation/low note
be/beeexist/flying insect
beach/beechsandy area/type of tree
beau/bowmale admirer/bend down or front
bell/bellemusical instrument/pretty girl
berth/birthsleeping place/arrival of baby
bite/bytenibble/unit of data
blew/blueforced air/colour
bloc/blockgroup of countries/cuboid
boar/borewild pig/boring person
board/boredflat object/weary
bough/bowbranch/bend down or front of ship
boy/buoymale child/floating marker
brake/breakslow down/shatter
breach/breechgap/part of gun
bread/bredfood made with yeast/brought up
bridle/bridalleather strap/to do with brides
broach/broochopen (a question)/piece of jewellery
browse/browslook through/hair above the eyes
but/buttalthough/water tank
buy/by/byepurchase/by means of/goodbye
cache/cashhidden hoard/notes and coins
cannon/canongun/collection of artistic works
canvas/canvasssail fabric/ask questions of
capital/capitolupper case or city/government building
carat/carrot/caret/karatweight/vegetable/arrow/share of gold
cast/castethrown/social class
cede/seedgive up/reproductive unit
ceiling/sealingroof/making watertight
cell/sellpart of body/exchange for money
cellar/sellerbasement/person selling
censor/sensorban (film etc)/measurement device
cent/scent/sentUS penny/perfume/dispatched
cents/scentsUS pennies/perfumes
cereal/serialbreakfast dish/TV show
cession/sessiongiving up/period of course
chased/chastepursued/like a virgin
cheap/cheepinexpensive/bird sound
chilly/chillicold/hot food
choir/quiregroup of singers/unit of paper
chord/cordgroup of notes/string
chute/shootdisposal passage/take shot at goal
coarse/courserough/track or route
colonel/kernelarmy rank/stone in fruit
complement/complimentgo well with/say something nice
coo/coupsound of dove/revolution
core/corpscentre/army unit
council/counselgoverning body/advice
cue/queuesnooker tool/line of people waiting
currant/currentdried fruit/electric flow
cymbal/symbolmusical instrument/icon
dam/damnriver barrier/damnation
days/daze24-hour periods/confuse
dear/deerexpensive/type of mammal
defused/diffusedmade safe/circulated
desert/dessertsandy zone/pudding
dew/do/duewater on grass/finish/owed
die/dyeexpire/colour (verb)
discreet/discretenot talkative/separate
doe/dough/dohfemale dear/unbaked bread/oh, no
draft/draughtpractise writing/on tap (eg beer)
dual/duelin two parts/fight with swords etc
earn/urn/ern or ernemake money/vase/type of bird
ewe/you/yewfemale sheep/2nd person/type of tree
faint/feintlose consciousness/fake attack
faun/fawnrural god/beige or young deer
faze/phase disturb/stage
feat/feetachievement/plural of foot
find/finedseek/told to pay money
fir/furtype of tree/animal hide
flair/flaretalent/bullet making bright light
flea/fleetype of insect/run away
flew/flu/fluepast tense of fly/influenza/chimney
flocks/phloxherds/type of plant
flour/floweringredient for bread/plant
for/four/foreto the benefit of/4/in front
friar/fryer monk/pan
gait/gateway of walking/door outside
gene/jeanDNA unit/trousers
gild/guildcover in gold/organisation
gilt/guiltcovered in gold/having done wrong
gored/gourdholed/fruit or water container
gorilla/guerrillatype of ape/freedom fighter
grease/Greecelubrication/a country
groan/grownmoan/past tense of grow
guessed/guestpast tense of guess/invitee
hair/harestrands growing on head/rabbit
hangar/hangerstorage for aircraft/hook in wardrobe
hay/heydried grass/oy
heal/heel/he’llmake well/back of foot/he will
hear/herelisten/in this place
heard/herdpast tense of hear/group of animals
heed/he’dpay attention to/he would or he had
hertz/hurtsfrequency unit/causes pain
hew/hue/Hughcut/colour/a name
higher/hiremore raised/rent
him/hymna pronoun/religious song
hoard/hordecollection/mass of people
hoarse/horserough (of voices)/an animal
hoes/hosegarden tools/tube
hold/holedkeep or carry/past tense of hole
holey/holy/whollywith holes/sacred/completely
hour/our60 minutes/a pronoun
humorous/humerusfunny/arm bone
idle/idollazy/religious statue
illicit/elicitillegal/draw out
it’s/itsit is/belonging to it
jam/jambfruit spread/door frame
kernel/colonelcore/army rank
knead/kneed/needmix dough/hit with knee/require
knight/nightwarrior/dark time
knit/nitfit together/egg of louse
knot/nottied rope/negative
know/no/Nohbe aware of/negative/type of drama
knows/noseis aware of/facial feature
laid/ladepast tense of lay/load ship
lain/lanepast participle of lay/alley
lay/leiplace/flower necklace
leach/leechleak/blood-sucking worm
lead/ledheavy metal/past tense of lead
leak/leekdrop out/vegetable
leased/leastpast tense of lease/superlative of less
lee/leashadow of wind/meadow
lessen/lessonmake less/teaching session
liar/lyreperson who lies/musical instrument
license/licencepermit (verb)/permission
lie/lyefalsehood/alkali solution
links/lynxconnections/wild cat
load/lodeput into/vein of metal in ground
locks/loxsecures/smoked salmon (American)
loot/lutemoney/musical instrument
made/maidcreated/young woman
main/mane/Mainechief/hair/state in USA
manner/manorway/lord’s house
marshal/martialarmy rank/to do with war
massed/mastbrought together/upright post on ship
maybe/may beperhaps/might be
meat/meet/metetype of food/get together/distribute
metal/mettleshiny material/spirit
might/mitemay/tiny spider
mince/mintsground beef/plural of mint
mind/minedbrain/dug up
miner/minor/mynahdigger/junior/type of bird
missed/mistpast tense of miss/fog
moan/mowngroan/past participle of mow
mode/mowedway/past tense of mow
muscle/musselpart of body/sea creature
mustard/musteredspicy dressing/broughted together
naval/navelto do with the navy/belly button
nay/neighno (dated)/sound of horse
none/nunnot one/female monk
oar/or/oreblade/alternatively/metal source
oh/owe/oah/have a debt of/oh (poetic)
pain/paneache/window panel
pair/pare/pearcouple/shave/type of fruit
palate/palette/palletpart of mouth/artist’s tray/platform
passed/pastpast tense of pass/in the past
patience/patientstolerance/people in hospital
pause/pawsbreak/animal hands and feet
peak/peek/piquesummit/look quickly/annoyance
peal/peelsound of bells/take skin off
pearl/purlprecious stone/knitting stitch
pedal/peddlefoot lever/sell
peer/pierlook carefully/jetty
per/purrfor each/sound of a cat
pi/pie3.14/dish topped with pastry
plain/planeunexciting/2D object
pleas/pleaserequests (noun)/if it pleases you
plum/plumbtype of fruit/measure water depth
pole/pollrod/election or survey
pore/pourconcentrate on/flow
pray/preytalk to God/victim
presence/presentsbeing somewhere/gifts
prince/printsson of monarch/printed photographs
principal/principlemain/rule of conduct
profit/prophetmoney made/religious seer
rack/wrackwire tray/shipwreck
rain/reign/reinwater from clouds/rule/control strap
raise/rays/razelift/plural of ray/destroy
rap/wraphit/pack up (eg a present)
rapped/rapt/wrappedpast tense of rap/spellbound/past tense of wrap
read/redpast tense of read/scarlet
read/reedstudy/type of plant
real/reelgenuine/cylinder for fishing line etc
reek/wreaksmell bad/cause
rest/wrestrelax/wrench away
retch/wretchvomit/poor soul
review/revuelook over/stage performance
right/rite/writecorrect/ritual/form Leopardtters
ring/wringsound of bell/squeeze out water
road/rode/rowedstreet/past tense of ride/past tense of row
roe/rowfemale deer/use oars
role/rollpart in play/type of bread
root/routepart of plant/roads to take
rose/rowstype of flower/tiers
rote/wroterepetition/past tense of write
rough/ruffcoarse/Elizabethan collar
rung/wrungpast tense of ring/squeezed water out
rye/wrycereal plant/mocking
sail/salecanvas propulsion/selling
scene/seensituation/past tense of see
scull/skullrow alone/head of skeleton
sea/seeocean/be aware of
seam/seemsewn connection/appear
seas/sees/seizeoceans/is aware of/grab
serf/surfagricultural worker/waves
sew/so/sowconnect with thread/thus/plant
shoe/shoofootwear/chase away
side/sighededge/past tense of sigh
sighs/sizebreathes out/dimensions
sleight/slightdeceptive skill/faint
soared/swordpast tense of soar/bladed weapon
some/suma few/total
son/sunmale child/star in the sky
staid/stayedunadventurous/past tense of stay
stair/starestep/look hard
stake/steakwooden post/joint of meat
stationary/stationerymotionless/writing materials
steal/steelrun off with/metal compound
step/steppestair/European plains
stile/stylefence steps/manner
straight/straitnot bending/narrow strip of water
suite/sweethotel rooms/sugary
summary/summerybrief account/to do with summer
surge/sergerush/type of cloth
tail/taleback end/story
taught/tautpast tense of teach/tight
tea/teemeal/golf ball holder
team/teemgroup of players/swarm
tern/turntype of bird/go round a corner
their/there/they’rebelonging to them/in that direction/they are
theirs/there’sthe one belonging to them/there is
threw/throughpast tense of throw/in and out of
thrown/thronepast participle of throw/royal chair
thyme/timetype of herb/progress of days or years
tic/tickhabit/mark (if correct)
tide/tiedflow of water/past tense of tie
to/too/twotowards/as well/2
toad/towedfrog/past tense of tow
toe/towpart of foot/pull
told/tolledpast tense of tell/rang
trussed/trustbound (with rope)/belief
vale/veilvalley/lace face covering (for brides etc)
vial/viletest tube/evil
wade/weighedwalk in water/past tense of weigh
wail/whalehowl/type of ocean mammal
waist/wastemiddle of body/use carelessly
wait/weightdelay or stay/mass
waive/wavegive up/breaker
ware/wear/wherepottery/put on clothes/which place
way/weigh/wheymanner/measure weight/part of milk
ways/weighsmanners/measures weight
we/weepronoun/urination or little
weak/weekfeeble/seven days
weather/whetherclimatic conditions/if
we’d/weedwe would or we had/unwanted plant
we’ll/wheelwe will or we shall/round component
we’ve/weavewe have/make cloth
while/wileas/ruse or cunning plan
whine/winewhimper/alcoholic grape drink
who’s/whosewho is or who has/of whom the
wood/wouldtree material/conditional marker
yoke/yolkpart of plough/yellow part of egg
yore/your/you’reformer times/belonging to you/you are
you’ll/Yuleyou will or you shall/Christmas

Circle theorems

This article explains circle theorems, including tangents, sectors, angles and proofs (with thanks to Revision Maths).

Isosceles Triangle

Two Radii and a chord make an isosceles triangle.

Perpendicular Chord Bisection

The perpendicular from the centre of a circle to a chord will always bisect the chord (split it into two equal lengths).

Angles Subtended on the Same Arc

Angles subtended on the same arc

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.

Angle in a Semi-Circle

angle in a semi-circle

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.


We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

Divide the triangle in two

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

Two isosceles triangles

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.


A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).

A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.

angle with a tangent

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

Tangents from an external point are equal in length

Angle at the Centre

Angle at the centre

The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.


You might have to be able to prove this fact:

proof diagram 1

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a
Similarly, ∠OXB = b

proof diagram 2

Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a
Similarly, ∠BOX = 180 – 2b
Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX
= 360 – (180 – 2a) – (180 – 2b)
= 2a + 2b = 2(a + b) = 2 ∠AXB

Alternate Segment Theorem

Alternate segment theorem

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.


You may have to be able to prove the alternate segment theorem:

proof of alternate segment theorem

We use facts about related angles

A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90
But OAC + x = 90, so ∠OAC + x = ∠OAC + y
Hence x = y

Cyclic Quadrilaterals

cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.

Area of Sector and Arc Length

A sector

If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360

In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360

How to become a private tutor


I’ve talked to a few people who wanted to become private tutors, so I thought I’d write down a few tips for anyone who’s interested.

How did I start out?

I started as a private tutor quite by accident. It was 2009, and I was finding it hard to get work as a freelance management consultant when I happened to read an article in the Telegraph called 10 Ways to Beat the Recession. The author mentioned a few ways of earning some extra cash, including becoming an extra on film sets – which I was already doing – and working as a private tutor. I’d never done any proper teaching before, although I was a golf coach, and I’d coached skiing a few times in the Alps, but I thought I’d sign up with a couple of agencies and see what happened. Within a week, I had two clients, and I’ve never looked back since!

What qualifications do I need?

The first and most important thing to say is that you don’t need any teaching qualifications! Yes, that’s right. You don’t need a PGCE, and you don’t need to have done any training as a teacher. As a private tutor, you are just that – private – so you don’t have to jump through all the Government hoops that a teacher in a state school would have to do. Obviously, potential clients want the best person to teach their child, so you need to show some sort of academic record, but that can be as little as a degree in English – which is what I had when I started. Admittedly, I went to Oxford, which probably counts for a lot with Russian billionaires (!), but you don’t need to have an Oxbridge degree to become a tutor. Far from it. However, what you probably will need is a criminal records check. This is just a piece of paper that certifies you haven’t been convicted of a criminal offence and was often known as a ‘CRB check’, although it’s now officially called an Enhanced Certificate from the Disclosure and Barring Service, or ‘DBS check’. You can’t apply for an ‘enhanced certificate’ yourself, but your tuition agency can help you. In fact, they may require you to have one and even to renew it every year or two. It costs around £18 and can take up to three months to arrive, so it’s worth applying as early as possible. Some agencies may charge up to £80 to make the application on your behalf, so be careful! You can find further information here.

What subjects can I teach?

You can teach whatever you like! Agencies will just ask you which subjects you offer and at what level, so you have complete freedom to choose. I focus on English and Maths, which are the most popular subjects, but that’s mostly led by demand from clients. They are the main subjects at 11+ level, so that’s what most people are looking for help with.

What age children can I teach?

Again, the choice is yours. I’ve taught students from as young as five to as old as 75, but the peak demand is at 11+ level, when the children are around 10 years old. I make it a rule that I’ll only teach a subject to a level that I’ve reached myself, such as GCSE or A-level, but clients sometimes take you by surprise. When I turned up to teach what I thought was going to be English to two boys, the nanny suddenly asked me to do Latin instead. When I said I hadn’t done any Latin since I was 15, she just said, “Oh, you’ll be fine…!”

What preparation do I need to do?

  • Research. One of the big attractions of tutoring for me is that the work is very enjoyable. I like teaching, and I like spending time with children, so it’s the perfect combination! The reason I stopped work as a management consultant was the constant stress, the persistent worry that I wasn’t up to the job, but teaching 10-year-olds never makes me feel like that. Whether it’s English or Maths, I’m confident in my ability to teach and never worry about being asked an impossible question. However, that doesn’t mean you can walk into your first lesson without doing any preparation at all. In my case, I wanted to teach English, so I needed to find out what kind of questions cropped up in 11+ and 13+ entrance exams and come up with a good method of answering them. Once I’d done that, I was ready. Maths was a bit easier, but I still looked through a few papers to make sure there was no risk of being blind-sided by something I’d forgotten how to do or had never studied. Whatever the subject you’re offering, I suggest you do the same.
  • Past papers. The other thing I needed to do was to find past papers to give to my pupils. That was a bit tricky in the early days until a kind parent gave me a collection of photocopied exams. After that, I carried a couple around with me to take to lessons, but it wasn’t a great solution, so I decided to create a website – this one. Over time, I collected dozens of past papers and wrote various articles on how to do different kinds of question in Maths, English and French. Now, I don’t have to carry around anything with me or spend time dictating notes. I can simply ask my pupils to look it up online. Setting up a website is pretty easy using WordPress or something similar, but you should feel free to use the resources on my past papers tab if you don’t want to go to the trouble yourself, and all my articles are available for free if you need them. The main ones I use for English are about doing comprehensions and writing stories, but there are plenty more. The website proved unexpectedly popular, and I had over 28,000 visitors last year! The other advantage is that it generated enough business for me not to need agencies any more. That means I can charge what I like, I don’t have to pay any commission, and I can have a direct relationship with all my clients without anybody acting as an intermediary – and often just getting in the way!
  • Business cards. I know it sounds a bit old-fashioned, but having business cards is very useful. If you’re just starting out, nobody knows your name, so paying a few quid to market your services is one of the best investments you can make. You never know when people will tell you they’re looking for a tutor, and it’s the easiest thing in the world to give them a business card. Even if you don’t have a website, it will at least tell them how to reach you, and you should get a lot more clients out of it.

How can I find work?

Tuition agencies are the best place to start, but there are different kinds. Some are online and simply require you to fill out a form for them to check and vet, but others ask you to go through an interview, either over the phone or in person. Either way, you need to put together a tailored CV that shows off your academic achievements and highlights any teaching experience you’ve had. This may not be very much at the beginning, but you simply need to show enough potential to get you through the door. Once you’ve shown enough aptitude and commitment to get accepted by a few agencies, you’ll rapidly build up your experience on the job.

Here is a list of the tuition agencies I’ve been in touch with, together with contact details where available. I’m based in London, so there is obviously a geographical bias there, but some of the agencies such as Fleet Tutors offer national coverage, and you can always search online for others in your local area.





A-Star Tuition astartuition@btinternet.com 01772 814739 astartuition.com
Approved Tutors approvedtutors.co.uk
Athena Tuition athenatuition.co.uk/
Beacon Tutors info@beacontutors.co.uk 020 8983 2158 beacontutors.co.uk/
Bespoke Tuition emma@bespoketuition.com 07732 371880 bespoketuition.com
Bigfoot Tutors tutors@bigfoottutors.com 020 7729 9004 bigfoottutors.com
Bright Young Things 07702 019194 brightyoungthingstuition.co.uk
Dulwich Tutors info@dulwichtutors.com 020 8653 3502 dulwichtutors.com
Enjoy Education kate@enjoyeducation.co.uk enjoyeducation.co.uk
Exam Confidence
First Tutors firsttutors.co.uk
Fleet Tutors 0845 644 5452 fleet-tutors.co.uk
Gabbitas gabbitas.com
Greater London Tutors 020 7727 5599 greaterlondontutors.com
Harrison Allen harrisonallen.co.uk
Holland Park Tuition recruitment@hollandparktuition.com 020 7034 0800 hollandparktuition.com
IPS Tutors info@ipstutors.co.uk 01509 265623 ipstutors.co.uk
Ivy Education ivyeducation.co.uk
Kensington & Chelsea Tutors tutors@kctutors.co.uk 020 7584 7987 kctutors.co.uk
Keystone Tutors enquiries@keystonetutors.com 020 7351 5908 keystonetutors.com
Kings Tutors emily@kingstutors.co.uk kingstutors.co.uk
Knightsbridge Tutors 07890 521390 knightsbridgetutors.co.uk
Laidlaw Education laidlaweducation.co.uk
Mentor & Sons andrei@mentorandsons.com 07861 680377 mentorandsons.com
Osborne Cawkwell enquiries@oc-ec.com 020 7584 5355 oc-ec.com
Personal Tutors admin@personal-tutor.co.uk personal-tutors.co.uk
Russell Education Group joe@russelleducationgroup.com n/a
Search Tutors searchtutors.co.uk
Select My Tutor info@selectmytutor.co.uk selectmytutor.co.uk
SGA Education s@sga-education.com sga-education.com
Simply Learning Tuition simplylearningtuition.co.uk
The Tutor Pages thetutorpages.com
Top Tutors 020 8349 2148 toptutors.co.uk
Tutor House info@tutorhouse.co.uk 020 7381 6253 tutorhouse.co.uk/
Tutor Hunt tutorhunt.com
Tutorfair tutorfair.com
Tutors International tutors-international.net
UK Tutors uktutors.com
Westminster Tutors exams@westminstertutors.co.uk 020 7584 1288 westminstertutors.co.uk
William Clarence Education steve@williamclarence.com 020 7412 8988 williamclarence.com
Winterwood  winterwoodtutors.co.uk

That’s obviously a long list, but, to give you an idea, I earned the most from Adrian Beckett (teacher training), Bespoke Tuition, Bonas MacFarlane, Harrison Allen, Keystone Tutors, Mentor & Sons, Personal Tutors and Shawcross Bligh.

Once you’ve been accepted by and started working for a few agencies, you’ll soon see the differences. Some offer higher rates, some the option to set your own rates, some provide a lot of work, some offer the best prospects of jobs abroad. It all depends what you’re looking for.

Where will the lessons take place?

When I first started tutoring, I had to cycle to all my clients. I put a limit of half an hour on my travel time, but it still took a lot of time and effort to get to my pupils. Fortunately, I’m now able to teach at my home, either in person or online using Skype and an electronic whiteboard, which means my effective hourly rate has gone up enormously. Travel is still a little bit of a problem for most tutors, though, and I certainly couldn’t have reached my pupils without having a bicycle. I didn’t have a car, and public transport wasn’t really an option in most cases. You just have to decide how far you’re prepared to go: the further it is, the more business you’ll get, but the longer it’ll take to get there and therefore the lower your effective hourly rate.

The other possibility, of course, is teaching abroad. I’ve been lucky enough to go on teaching assignments in Belarus, Greece, Hong Kong, Kenya, Russia, Switzerland and Turkey, and it’s a great way to see the world. The clients can sometimes be a little bit difficult, and the children can sometimes behave like spoiled brats (!), but staying with a great client in a sunny getaway overseas can be a wonderful experience. The only reason I don’t apply for more foreign postings is that I don’t want to let down my existing clients – going away for three weeks just before the 11+ exams in January would NOT go down well!

When will the lessons take place?

If you’re teaching children, lessons will usually be in the after-school slot between 1600 and 2000 or at weekends. That does limit the amount of hours you can teach, but it’s up to you how much you want to work. I used to work seven days a week, but I eventually gave myself a day off and then another, so I now work Sundays to Thursdays with Friday and Saturday off. During the holidays, you lose a lot of regular clients when they disappear to the Maldives or somewhere for six weeks (!), but others might ask for an intensive sequence of lessons to take advantage of the extra time available, and there’s obviously a greater chance of a foreign assignment. All that means that the work is very seasonal, so you should expect your earnings to go up and down a bit and plan your finances accordingly.

What should I do during the lesson?

I generally teach from past papers, so I ask pupils to do a past paper for their homework and then mark it during the following lesson. ‘Marking’ means marking the questions, obviously, but it also means ‘filling in the gaps’ in the pupil’s knowledge. If he or she is obviously struggling with something, it’s worth spending a few minutes explaining the topic and asking a few practice questions. I’ve written a few articles on common problem areas in English and Maths, such as commas and negative numbers, so I often go through one of those and ask the pupil’s parents to print it out and put it in a binder. After a few weeks, that collection of notes gradually turns into a ready-made revision guide for the exams.

If the parents want you to work on specific topics, that’s also possible. For example, one mother wanted to help her son with ratios, so she printed out dozens of past papers and circled the ratio questions for him to do. He soon got the knack!

I approach English in a slightly different way to begin with. There are two types of question in the 11+, comprehensions and creative writing, so I generally spend the first lesson teaching pupils how to do one of those. I go through my article on the subject online and then ask them to answer a practice question by following the procedure I’ve outlined. They usually finish it off for their homework. After a few weeks of stories or comprehensions, I’ll switch to the other topic and do the same with that. I also ask pupils to write down any new words or words they get wrong in a vocabulary book because building vocabulary is very important for any type of English exam (and also for Verbal Reasoning). I ask them to fold the pages over in the middle so that they can put the words on the left and the meanings on the right (if necessary). Every few weeks, I can then give them a spelling test. If they can spell the words correctly and tell me what they mean, they can tick them off in their vocab book. Once they’ve ticked off a whole page of words, they can tick that off, too! I usually try to reinforce the learning of words by asking pupils to tell me a story using as many words as possible from their spelling test. It can be a familiar fairy story or something they make up, but it just helps to move the words from the ‘passive’ memory to the ‘active memory’, meaning that they actually know how to use them themselves rather than just understand them when they see them on the page.

What homework should I set?

Most children who have private lessons have pretty busy schedules, so I don’t want to overburden them. I generally set one exercise that takes around 30-45 minutes. That might be a Maths paper or an English comprehension or story, but it obviously depends on the subject and the level. Just make sure that the student writes down what needs to be done – a lot of them forget! You should also make a note in your diary yourself, just so that you can check at the start of the next lesson if the work has been done.

What feedback should I give the parents?

I generally have a quick chat with the mother or father (or nanny) after the lesson to report on what we did during the lesson, what problems the child had and what homework I’ve set. This is also a good time to make any changes to the schedule, for instance if the family goes on holiday. If that’s not possible, I’ll email the client with a ‘lesson report’. Some agencies such as Bonas MacFarlane make this a part of their timesheet system.

How much will I get paid?

When I first started, I had absolutely no idea how much I was worth, and I ended up charging only £10 an hour, which is not much more than I pay my cleaner! Fortunately, a horrified friend pointed out that it should be ‘at least’ £35 an hour, and I upped my rates immediately. I now charge £60 an hour for private lessons, whether online or in person. Unfortunately, some agencies such as Fleet Tutors don’t allow you to set your own rates, so that’s one thing to bear in mind when deciding which agencies to work with. However, they did provide me with quite a bit of work when I first started, so it’s swings and roundabouts. The pay scale often varies depending on the age of the student and the level taught, so you’ll probably earn more for teaching older students at GCSE level or above if the agency sets the prices. If you have any private clients, you can obviously set whatever rate you like, depending on where you live, the age of your pupils, whether lessons are online or in person and so on. Personally, I only have one rate (although I used to charge an extra £5 for teaching two pupils at the same time), and I raise it by £5 every year to allow for inflation and extra demand. Tutoring is more and more popular than ever these days, and I read somewhere that over half of pupils in London have private lessons over the course of their school careers, so don’t sell yourself short! You should be able to make around £25,000 a year, which is not bad going for a couple of hours’ work a day!

Foreign jobs are a little different, and there is a ‘standard’ rate of around £800 a week including expenses. That means your flights and accommodation are all covered, and you can even earn a bit more on the side if you decide to rent out your home on Airbnb while you’re away! When it comes to day-to-day expenses such as taxis and food and drink, it’s important to negotiate that with the agency before accepting the job. It’s no good complaining about having to live in the client’s house and buy your own lunches when you’re in Moscow or Bratislava! It can be a dream job, but just make sure you look at it from every angle:

  • What subjects will I be teaching?
  • How many hours will I have to teach?
  • How many days off will I get per week?
  • Where will the lessons take place?
  • How do I get to and from my accommodation?
  • How long is the assignment? (I refuse anything more than three months.)
  • Where will I be staying? (NEVER at the client’s house!)
  • How old are the children?
  • Will I have any other responsibilities (eg ferrying the children to and from school)?
  • Do I need a visa?
  • What is the weekly rate?
  • What expenses are included (eg flights, accommodation, taxis, food, drink)?

How do I get paid?

Most agencies ask for a timesheet and pay their tutors monthly via BACS payments directly into their bank accounts. That’s a bit annoying from a cash flow point of view, but there’s not much you can do about it – other than using a different agency. When it comes to private clients, I generally ask for cash after the lesson, but it’s even more convenient if they can pay via standing order – as long as you can trust them! I once let a client rack up over £600 in fees because he tended to pay in big lump sums every few weeks, but then his business folded, and I had to use a Government website to try and chase him up. Fortunately, his wife saw the email and paid my bill, but it took months to sort out. Normally, though, the worst that happens is that a client just doesn’t have the right change and promises to pay the following week, so you just need to keep track of who owes what.

I hope all this helps. Good luck!



Nothing makes the heart of a reluctant mathematician sink like an algebra question.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…

Gathering terms

X’s and y’s look a bit meaningless, but that’s the point. They can stand for anything. The simplest form of question you’ll have to answer is one that involves gathering your terms. That just means counting how many variables or unknowns you have (like x and y). I like to think of them as pieces of fruit, so an expression like…

2x + 3y – x + y

…just means ‘take away one apple from two apples and add one banana to three more bananas’. That leaves you with one apple and four bananas, or…

x + 4y

Here are a few practice questions:

  1. 3x + 4y – 2x + y
  2. 2m + 3n – m + 3n
  3. p + 2q + 3p – 3q
  4. 2a – 4b + a + 4b
  5. x + y – 2x + 2y

Multiplying out brackets

This is one of the commonest types of question. All you need to do is write down the same expression without the brackets. To take a simple example:

2(x + 3)

In this case, all you need to do is multiply everything inside the brackets by the number outside, which is 2, but what do we do about the ‘+’ sign? We could just multiply 2 by x, write down ‘+’ and then multiply 2 by 3:

2x + 6

However, that gets us into trouble if we have to subtract one expression in brackets from another (see below for explanation) – so it’s better to think of the ‘+’ sign as belonging to the 3. In other words, you multiply 2 by x and then 2 by +3. Once you’ve done that, you just convert the ‘+’ sign back to an operator. It gives exactly the same result, but it will work ALL the time rather than just with simple sums!

Here are a few practice questions:

  1. 2(a + 5)
  2. 3(y + 2)
  3. 6(3 + b)
  4. 3(a – 3)
  5. 4(3 – p)

Solving for x

Another common type of question involves finding out what x stands for (or y or z or any other letter). The easiest way to look at this kind of equation is using fruit again. In the old days, scales in a grocery shop sometimes had a bowl on one side and a place to put weights on the other. To weigh fruit, you just needed to make sure that the weights and the fruit balanced and then add up all the weights. The point is that every equation always has to balance – the very word ‘equation’ comes from ‘equal’ – so you have to make sure that anything you do to one side you also have to do to the other.

There are three main types of operation you need to do in the following order:

  1. Multiplying out any brackets
  2. Adding or subtracting
  3. Multiplying or dividing

Once you’ve multiplied out any brackets (see above), what you want to do is to simplify the equation by removing one expression at a time until you end up with something that says x = The Answer. It’s easier to start with adding and subtracting and then multiply or divide afterwards (followed by any square roots). To take the same example as before:

2(x + 3) = 8

Multiplying out the brackets gives us:

2x + 6 = 8

Subtracting 6 from both sides gives us:

2x = 2

Dividing both sides by 2 gives us the final answer:

x = 1


Here are a few practice questions:

  1. b + 5 = 9
  2. 3y = 9
  3. 6(4 + c) = 36
  4. 3(a – 2) = 24
  5. 4(3 – p) = -8

Multiplying two expressions in brackets (‘FOIL’ method)

When you have to multiply something in brackets by something else in brackets, you should use what’s called the ‘FOIL’ method. FOIL is an acronym that stands for:


This is simply a good way to remember the order in which to multiply the terms, so we start with the first terms in each bracket, then move on to the outside terms in the whole expression, then the terms in the middle and finally the last terms in each bracket. Just make sure that you use the same trick we saw earlier, combining the operators with the numbers and letters before multiplying them together. For example:

(a + 1)(a + 2)

First we multiply the first terms in each bracket:

a x a

…then the outside terms:

a x +2

…then the inside terms:

+1 x a

…and finally the last terms in each bracket:

+1 x +2

Put it all together and simplify:

(a + 1)(a + 2)

= a² + 2a + a + 2

=a² + 3a + 2

Here are a few practice questions:

  1. (a + 1)(b + 2)
  2. (a – 1)(a + 2)
  3. (b + 1)(a – 2)
  4. (p – 1)(q + 2)
  5. (y + 1)(y – 3)

Factorising quadratics (‘product and sum’ method)

This is just the opposite of multiplying two expressions in brackets. Normally, factorisation involves finding the Highest Common Factor (or HCF) and putting that outside a set of brackets containing the rest of the terms, but some expressions can’t be solved that way, eg a² + 3a + 2 (from the previous example). There is no combination of numbers and/or letters that goes evenly into a², 3a and 2, so we have to factorise using two sets of brackets. To do this, we use the ‘product and sum’ method. This simply means that we need to find a pair of numbers whose product equals the last number and whose sum equals the multiple of a. In this case, it’s 1 and 2 as +1 x +2 = +2 and +1 + +2 = +3. The first term in each bracket is just going to be a as a x a = a². Hence, factorising a² + 3a + 2 gives (a + 1)(a + 2). You can check it by using the FOIL method (see above) to multiply out the brackets:

(a + 1)(a + 2)

= a² + 2a + a + 2

=a² + 3a + 2

Subtracting one expression from another*

Here’s the reason why we don’t just write down operators as we come across them. Here’s a simple expression we need to simplify:

20 – 4(x – 3) = 16

If we use the ‘wrong’ method, then we get the following answer:

20 – 4(x – 3) = 16

20 – 4x – 12 = 16

8 – 4x = 16

4x = -8

x = -2

Now, if we plug our answer for x back into the original equation, it doesn’t balance:

20 – 4(-2 – 3) = 16

20 – 4 x -5 = 16

20 – -20 = 16

40 = 16!!

That’s why we have to use the other method, treating the operator as a negative or positive sign to be added to the number before we multiply it by whatever’s outside the brackets:

20 – 4(x – 3) = 16

20 – 4x + 12 = 16

32 – 4x = 16

4x = 16

x = 4

That makes much more sense, as we can see:

20 – 4(4 – 3) = 16

20 – 4 x 1 = 16

20 – 4 = 16

16 = 16

Thank Goodness for that!

Here are a few practice questions:

  1. 30 – 3(p – 1) = 0
  2. 20 – 3(a – 3) = 5
  3. 12 – 4(x – 2) = 4
  4. 24 – 6(x – 3) = 6
  5. 0 – 6(x – 2) = -12

Other tips to remember

  • If you have just one variable, leave out the number 1, eg 1x is just written as x.
  • When you have to multiply a number by a letter, leave out the times sign, eg 2 x p is written as 2p.
  • The squared symbol only relates to the number or letter immediately before it, eg 3m² means 3 x m x m, NOT (3 x m) x (3 x m).


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…

Here be ratios...!

Here be ratios…!

A ratio is just a model of the real world, shown in the lowest terms, but answering ratio questions can be just as scary as meeting dragons if you don’t know what you’re doing. The key to understanding ratios is to work out the scale factor. This is just like the scale on a map. If a map is drawn to a scale of 1:100,000, for instance, you know that 1cm on the map is the same as 100,000cm (or 1km) in the real world. To convert distances on the map into distances in the real world, you just need to multiply by the scale factor, which is 100,000 in this case. (You can also go the other way – from the real world to the map – by dividing by the scale factor instead.)

To work out the scale factor in a Maths question, you need to know the matching quantities in the real world and in the model (or ratio). Once you know those two numbers, you can simply divide the one in the real world by the one in the ratio to get the scale factor. For example:

If Tom and Katie have 32 marbles between them in the ratio 3:1, how many marbles does Tom have?

To answer this question, here are the steps to take:

  1. Work out the scale factor. The total number of marbles in the real world is 32, and the total in the ratio can be found by adding the amounts for both Tom and Katie, which means 3 + 1 = 4. Dividing the real world total by the ratio total gives 32 ÷ 4 = 8, so the scale factor is 8.
  2. Multiply the number you want in the ratio by the scale factor. If Tom’s share of the marbles in the ratio is 3, then he has 3 x 8 = 24 marbles.

The matching numbers in the real world and the ratio are sometimes the totals and sometimes the individual shares, but it doesn’t matter what they are. All you need to do is find the same quantity in both places and divide the real world version by the ratio version to get the scale factor. Once you’ve done that, you can multiply any of the ratio numbers to get to the real world number (or divide any real world number to get to the ratio number). Different questions might put the problem in different ways, but the principle is the same.

One complication might be having two ratios that overlap. In that case you need to turn them into just one ratio that includes all three quantities and answer the question as you normally would. For example:

If there are 30 black sheep, and the ratio of black to brown sheep is 3:2, and the ratio of brown to white sheep is 5:4, how many white sheep are there?

This is a bit more complicated, but the basic steps are the same once you’ve found out the ratio for all three kinds of sheep. To do this, we need to link the two ratios together somehow, but all the numbers are different, so how do we do it? The answer is the same as for adding fractions with different denominators (or for solving the harder types of simultaneous equations, for that matter): we just need to multiply them together. If we were adding fifths and halves, we would multiply the denominators together to convert them both into tenths. Here, the type of sheep that is in both ratios is the brown one, so we simply have to make sure the numbers of brown sheep in each ratio (2 and 5) are the same by multiplying them together (to give 10). Once we’ve done that, we can combine the two ratios into one and answer the question. Here goes:

  • Ratio of black sheep to brown sheep = 3:2

Multiply by 5

  • Ratio of black sheep to brown sheep = 15:10
  • Ratio of brown to white sheep = 5:4

Multiply by 2

  • Ratio of brown to white sheep = 10:8
  • Therefore, ratio of black sheep to brown sheep to white sheep = 15:10:8

Now that we have just one ratio, we can answer the question by following exactly the same steps as before:

  1. Work out the scale factor. The total number of black sheep in the real world is 30, and the total in the ratio is 15. Dividing the real world total by the ratio total gives 30 ÷ 15 = 2, so the scale factor is 2.
  2. Multiply the number you want in the ratio by the scale factor. If the number of white sheep in the ratio is 8, then there are 8 x 2 = 16 white sheep.


Here are a few practice questions:

  1. One hundred paintings have to be selected for an art exhibition. If the ratio of amateur paintings to professional paintings has to be 2:3, how many amateur paintings and professional paintings have to be selected?
  2. The ratio of brown rats to black rats is 3:2. If there are 16 black rats, how many brown rats are there?
  3. Peter has 20 blue pens. How many red pens must he buy if the ratio of blue to red pens has to be 2:3?
  4. There are 35 children in a class and 15 are boys. What is the ratio of boys to girls?
  5. There are 15 girls and 12 boys in a class. What is the ratio of girls to boys? Give your answer in its simplest form.
  6. A newspaper includes 12 pages of sport and 8 pages of TV. What is the ratio of sport to TV? Give your answer in its simplest form.
  7. Anna has 75p, and Fiona has £1.20. What is the ratio of Anna’s money to Fiona’s money in its simplest form?
  8. Sam does a scale drawing of his kitchen. He uses a scale of 1:100. He measures the length of the kitchen as 5.9m. How long is the kitchen on the scale drawing? Give your answer in mm.
  9. A recipe to make lasagne for 6 people uses 300 grams of minced beef. How much minced beef would be needed to serve 8 people?
  10. A recipe for flapjacks requires 240g of oats. This makes 18 flapjacks. What quantity of oats is needed to make 24 flapjacks?
  11. Amit is 12 years old. His brother, Arun, is 9. Their grandfather gives them £140, which is to be divided between them in the ratio of their ages. How much does each of them get?
  12. The angles in a triangle are in the ratio 1:2:9. Find the size of the largest angle.
  13. In a certain town, the ratio of left-handed people to right-handed people is 2:9. How many right-handed people would you expect to find in a group of 132 people?
  14. Twelve pencils cost 72p. Find the cost of 30 pencils.
  15. Jenny buys 15 felt-tip pens. It costs her £2.85. How much would 20 pens have cost?
  16. If three apples cost 45p, how much would five apples cost?
  17. Sam is 16 years old. His sister is 24 years old. What’s the ratio of Sam’s age to his sister’s age? Give your answer in its simplest form.
  18. A map scale is 1:20000. A distance on the map is measured to be 5.6cm. What’s the actual distance in real life? Give your answer in metres.
  19. A recipe for vegetable curry needs 300 grams of rice, and it feeds 4 people. How much rice would be needed for 7 people?
  20. £60 is to be divided between Brian and Kate in the ratio 2:3. How much does Kate get?


Teaching Greek children is like watching France play rugby: you never know what you’re going to get…

The Stoa of Attalos marble colonnade and ceiling

Stoa of Attalos: the Athenian version of the local mall


I just spent two weeks in Greece preparing a Greek boy and his twin sisters for 10+ and 12+ entrance examinations at a school in England. Highlights included spending a long, sunny weekend at a holiday home in Lagonissi, spending another long, sunny weekend skiing near Delphi – I wonder if the oracle saw that one coming! – and seeing the Parthenon every day from my hotel balcony.

Political refugees take many forms, but, personally, I prefer shipping magnates fleeing with their adorable (if strong-willed) families from Communist governments in the Mediterranean…