Tag Archives: entrance exams

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. Even if it’s easy enough to do in your head, you still have to write down the sum on paper. That way, the examiner knows that you didn’t just guess!

Here are the basic operations:

Addition

The standard way to add numbers is the ‘column method’.

  • Write down the numbers one on top of the other (however many there are), with two lines under them and a plus sign on the left.
  • Add the first column of numbers on the right and put the answer between the lines.
    • If the total is more than 9, ‘carry’ the tens by putting that number in small handwriting under the next space on the answer line.
  • Add the next column of numbers working from the right and put the answer between the lines, adding any numbers below the line that have been carried.
  • If you get to the final column of numbers and the total is more than 9, you can write both digits on the answer line.
  • If you have more than two columns of numbers and the total is more than 9, you’ll have to ‘carry’ any tens again by putting that number in small handwriting under the next space on the answer line.
  • You can then finish off as normal.

Notes:

  • You don’t need the second line if you don’t want to use it.
  • You can also choose to put the carried numbers above the top line of the sum, but that gets a bit messy if you’re doing long multiplication, so it’s best to get into the habit of using this method.

Sample questions:

Have a go at these questions. Don’t just do them in your head. That’s too easy! Make sure you show your working – just as you’d have to do in an exam.

  1. 8 + 5
  2. 17 + 12
  3. 23 + 19
  4. 77 + 88
  5. 127 + 899

Subtraction

The standard way to subtract one number from another is again the ‘column method’, but this time it’s slightly different. For a start, you can only use this method with two numbers (not three or more), and you can’t use it for negative numbers.

  • Write down the two numbers one on top of the other, with the bigger one on top, the usual two lines under them and a minus sign on the left.
  • Working from the right, take away the first digit in the second number from the first digit in the first and write the answer on the answer line.
    • If you can’t do it because the digit on the top row is too small, you’ll have to ‘borrow’ a 10 from the digit in the next column.
      • Place a 1 above and to the left of the top right-hand digit to make a new number, in this case 12.
      • Cross out the digit you’re borrowing from, subtract 1 and write the new digit above and to the left of the old one.
      • You can now subtract as normal, so 12 – 7 = 5 in this case.
  • Working from the right, subtract the next digit in the bottom number from the next digit in the top number and put the answer between the lines.
  • Repeat this step until you’ve finished the sum.
    • Note that in this case you have to borrow 1 from the 2, leaving 1, and then borrow 1 from the 4, writing it next to the 1 so it makes 11. It may look like you’re borrowing 11, but you’re not. You’ve just had to write the two 1s next to each other.

If you can’t borrow from a digit because it’s a zero, just cross it out, write 9 above and to the left and borrow from the next digit to the left. If that’s a zero, too, just do the same again until you reach one that’s not zero.

Notes:

  • You don’t need the second line if you don’t want to use it.
  • If the answer to the sum in the last column on the left is zero, you don’t need to write it down, so your answer should be 17, say, not 017.
  • You don’t need to put commas in numbers that are more than 1,000.
  • You could cross out the numbers from top left to bottom right instead, but that leaves less room to write any little numbers above and to the left (where they have to go), so it’s best to get into the habit of using this method.

Sample questions:

Have a go at these questions. Don’t just do them in your head. That’s too easy! Make sure you show your working – just as you’d have to do in an exam.

  1. 8 – 5
  2. 17 – 12
  3. 43 – 19
  4. 770 – 681
  5. 107 – 89

Multiplication (or short multiplication)

This is short multiplication, which is meant for multiplying one number by another that’s in our times tables (up to 12). 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 single-digit number on the bottom, two lines underneath and a times sign on the left.
  • Multiply the last digit of the top number by the bottom number and put the answer between the lines.
    • If the total is more than 9, ‘carry’ the tens by putting that number in small handwriting under the next space on the answer line.
  • Working from the right, multiply the next digit of the top number by the bottom number, adding any number below the answer line.
    • As with addition, if you get to the final column of numbers and the total is more than 9, you can write both digits on the answer line.

Notes:

  • You don’t need the second line if you don’t want to use it.
  • You can also choose to put the carried numbers above the top line of the sum, but that gets a bit messy if you’re doing long multiplication, so it’s best to get into the habit of using this method.

Sample questions:

Have a go at these questions. Don’t just do them in your head. That’s too easy! Make sure you show your working – just as you’d have to do in an exam.

  1. 21 x 3
  2. 17 x 4
  3. 23 x 6
  4. 77 x 8
  5. 127 x 9

Division (or short division, or the ‘bus stop’ method)

This is short division, which is meant for dividing one number by another that’s in your times tables (up to 12). If you want to divide by a higher number, you need to use long division (see my article here). It’s called the ‘bus stop’ method because the two lines look a bit like the area where a bus pulls in at a bus stop.

  • Write down the number you’re dividing (the ‘dividend’), draw the ‘bus stop’ shape around it so that all the digits are covered and then write the number you’re dividing by (the ‘divisor’) on the left.
  • Try to divide the first digit of the dividend by the divisor. If it goes in exactly, write the answer on the answer line above the first digit of the dividend.
  • If it goes in, but there’s a remainder, write the answer on the answer line above the first digit of the dividend and then write the remainder above and to the left of the next digit in the dividend.
  • If it doesn’t go, then make a number out of the first two digits of the dividend and divide that number by the divisor, adding any remainder above and to the left of the next digit.
  • Repeat this process for each of the remaining digits, using any remainders to make a new number with the next digit.
  • If you divide one number by another in the middle of the dividend and it doesn’t go, then just put a zero on the answer line and combine the digit with the next one.

Notes:

  • If you have a remainder at the end of the sum, you can either show it as a remainder or you can put a decimal point above and below the line, add a zero to the dividend and carry on until you have no remainder left.
    • If the remainder keeps going, it’s likely to repeat the same digits over and over again. This is called a ‘recurring decimal’. Once you spot the pattern, you can stop doing the sum. Just put a dot over the digit that’s repeating or – if there’s more than one – put a dot over the first and last digit in the pattern.
  • If your handwriting is a bit messy, make sure you make the numbers quite large with a bit of space between them so that you can fit everything in!

Sample questions:

Have a go at these questions. Don’t just do them in your head. That’s too easy! Make sure you show your working – just as you’d have to do in an exam.

  1. 36 ÷ 3
  2. 172 ÷ 4
  3. 222 ÷ 6
  4. 816 ÷ 8
  5. 126 ÷ 9

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.

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:

SpellingsMeanings
a lot/allotmuch/distribute
ad/addnotice/put together
ads/adds/adzenotices/puts together/type of axe
aid/aidehelp/assistant
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
assent/ascentagreement/rise
assistance/assistantshelp/helper
ate/eightconsumed/8
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)
ball/bawlsphere/
band/bannedgroup/illegal
bard/barredpoet/banned
bare/bearnaked/tolerate or grizzly
baron/barrenlord/arid
base/bassfoundation/low note
be/beeexist/flying insect
beach/beechsandy area/type of tree
beat/beethit/beetroot
beau/bowmale admirer/bend down or front
bell/bellemusical instrument/pretty girl
berry/buryfruit/inter
berth/birthsleeping place/arrival of baby
billed/buildinvoiced/construct
bite/bytenibble/unit of data
blew/blueforced air/colour
bloc/blockgroup of countries/cuboid
boar/borewild pig/boring person
board/boredflat object/weary
boarder/borderlodger/edge
bode/bowedimply/curved
bolder/boulderbraver/rock
born/bornecreated/tolerated
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
brewed/broodfermented/family
brews/bruiseferments/scar
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
callous/callusheartless/blister
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
carol/carrelsong/cubicle
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
chance/chantsluck/songs
chased/chastepursued/like a virgin
cheap/cheepinexpensive/bird sound
chews/choosenibbles/select
chilly/chillicold/hot food
choir/quiregroup of singers/unit of paper
chord/cordgroup of notes/string
chute/shootdisposal passage/take shot at goal
cite/sight/sitequote/seeing/location
clause/clawsparagraph/talons
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
correspondence/correspondentsletters/letter-writers
council/counselgoverning body/advice
councillor/counsellorgovernor/advisor
creak/creekwhine/stream
crews/cruiseteams/voyage
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
done/dunfinished/grey-brown
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
fair/farejust/food
fated/feteddestined/celebrated
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
fort/fortecastle/speciality
forth/fourthforwards/4th
foul/fowldisgusting/birds
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
grate/greatfireplace/grand
grease/Greecelubrication/a country
groan/grownmoan/past tense of grow
guessed/guestpast tense of guess/invitee
hail/halecelebrate/healthy
hair/harestrands growing on head/rabbit
hall/haulroom/pull
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
hi/highhello/raised
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
hole/wholespace/entire
holey/holy/whollywith holes/sacred/completely
hour/our60 minutes/a pronoun
humorous/humerusfunny/arm bone
idle/idollazy/religious statue
illicit/elicitillegal/draw out
in/inninside/hotel
instance/instantsexample/moments
intense/intentsfierce/purposes
it’s/itsit is/belonging to it
jam/jambfruit spread/door frame
kernel/colonelcore/army rank
knap/napcrest/doze
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
levee/levyembankment/tax
liar/lyreperson who lies/musical instrument
license/licencepermit (verb)/permission
lichen/likenmould/compare
lie/lyefalsehood/alkali solution
links/lynxconnections/wild cat
load/lodeput into/vein of metal in ground
loan/lonelending/single
locks/loxsecures/smoked salmon (American)
loot/lutemoney/musical instrument
made/maidcreated/young woman
mail/malepost/masculine
main/mane/Mainechief/hair/state in USA
maize/mazecorn/labyrinth
manner/manorway/lord’s house
mantel/mantlementalpiece/coat
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
medal/meddleaward/interfere
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
moose/mousseelk/foam
morn/mournmorning/regret
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
ode/owedpoem/due
oh/owe/oah/have a debt of/oh (poetic)
overseas/overseesforeign/manages
pail/palebucket/faint
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
pea/peevegetable/urinate
peace/pieceharmony/bit
peak/peek/piquesummit/look quickly/annoyance
peal/peelsound of bells/take skin off
pearl/purlprecious stone/knitting stitch
pedal/peddlefoot lever/sell
pedalled/peddledcycled/sold
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
practice/practiserehearsal/rehearse
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
shear/sheercut/complete
shoe/shoofootwear/chase away
side/sighededge/past tense of sigh
sighs/sizebreathes out/dimensions
slay/sleighkill/sled
sleight/slightdeceptive skill/faint
soar/sorerise/painful
soared/swordpast tense of soar/bladed weapon
sole/soulonly/spirit
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
tacks/taxnails/levy
tail/taleback end/story
taught/tautpast tense of teach/tight
tea/teemeal/golf ball holder
team/teemgroup of players/swarm
tear/tierteardrop/row
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
vain/vane/veinproud/fin/artery
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
wet/whetliquid/sharpen
we’ve/weavewe have/make cloth
which/witchpronoun/wizard
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

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…

A bloke called Bob (actually Robert Plutchik) thought that people only ever felt eight different emotions:

His list is shown in this ‘wheel of emotions’. The basic eight feelings are:

  • Ecstasy
  • Admiration
  • Terror
  • Amazement
  • Grief
  • Loathing
  • Rage
  • Vigilance

If we had a think about all the adjectives that are associated with these categories (and sub-categories), we might come up with a list like this one:

  • Ecstasy
    crazy
    delirious
    ecstatic
    elated
    enthusiastic
    euphoric
    fervent
    glad
    happy
    joyful
    joyous
    mad
    overjoyed
    rapturous
    rhapsodic
    serene
    thrilled
    upbeat
  • Admiration
    accepting
    admiring
    adoring
    appreciative
    loving
    respectful
    trustful
    trusting
  • Terror
    afraid
    aghast
    alarmed
    apprehensive
    awed
    frightened
    frozen
    scared
    submissive
    terrified
  • Amazement
    amazed
    astonished
    astounded
    awe-struck
    bewildered
    dazed
    distracted
    dumbfounded
    flabbergasted
    impressed
    perplexed
    shocked
    staggered
    startled
    stunned
    surprised
    unprepared
  • Grief
    bitter
    grief-stricken
    grieving
    heart-broken
    melancholy
    mournful
    pensive
    pessimistic
    sad
    somber
    sorrowful
    sorry
    unhappy
    wistful
  • Loathing
    appalled
    bored
    disapproving
    disgusted
    outraged
    queasy
    tired
    weary
  • Rage
    angry
    aggressive
    annoyed
    contemptuous
    enraged
    exasperated
    furious
    heated
    impassioned
    indignant
    irate
    irritable
    irritated
    offended
    resentful
    sullen
    uptight
  • Vigilance
    anxious
    aware
    cautious
    circumspect
    expectant
    interested
    keen
    observant
    optimistic
    vigilant
    wary