Year 5 cheat sheet: what to know by heart
One page of the facts, formulas and shortcuts a Year 5 student sitting a selective or scholarship paper should be able to recall without working them out.
How to use this page
Everything here is meant to be recalled, not worked out. In a selective or scholarship paper you have under a minute a question, and a child who has to calculate that a quarter is 25% has already spent that minute.
Pick one table a day. Cover a column, say the row out loud, then check. Ten minutes a day for a fortnight will do more than an hour the night before.
Fractions, decimals and percentages
The single most useful table on this page. A dot after a decimal means the digits repeat forever.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33⅓% |
| 2/3 | 0.666… | 66⅔% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/6 | 0.1666… | 16⅔% |
| 5/6 | 0.8333… | 83⅓% |
| 1/7 | 0.142857… | ≈ 14.3% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/9 | 0.111… | 11.1% |
| 1/10 | 0.1 | 10% |
| 1/12 | 0.08333… | 8⅓% |
| 1/20 | 0.05 | 5% |
| 1/25 | 0.04 | 4% |
| 1/50 | 0.02 | 2% |
Two patterns worth spotting: the eighths go up in 0.125, and the ninths are just the digit repeated — 2/9 is 0.222…, 5/9 is 0.555….
Worked example — which is bigger, 3/8 or 0.4? The table says 3/8 is 0.375, and 0.375 is less than 0.4, so 0.4 wins. Turning both into decimals is almost always quicker than finding a common denominator.
Percentages without a calculator
Build every percentage out of 10%, 1% and halving.
| Want | Do this | Example: of 240 |
|---|---|---|
| 10% | divide by 10 | 24 |
| 1% | divide by 100 | 2.4 |
| 5% | 10%, then halve | 12 |
| 15% | 10% + 5% | 36 |
| 20% | divide by 5 | 48 |
| 25% | divide by 4 | 60 |
| 12.5% | divide by 8 | 30 |
| 50% | halve | 120 |
| 75% | half + quarter | 180 |
| 33⅓% | divide by 3 | 80 |
| 66⅔% | divide by 3, double | 160 |
Percentage change = change ÷ original × 100. The original is always the number you started with, which is the half of it most marks are lost on.
Worked example — a $60 jacket with 15% off. 10% of 60 is 6, and half of that is 3, so the saving is $9 and the jacket costs $51.
The arch method: moving the decimal point
Multiplying and dividing by ten, a hundred or a thousand never needs working out. Count the zeros and hop the point that many places — right to multiply, left to divide. Fill any empty place with a zero.
| Do this | The point hops | 3.47 becomes |
|---|---|---|
| × 10 | 1 place right | 34.7 |
| × 100 | 2 places right | 347 |
| × 1000 | 3 places right | 3470 |
| ÷ 10 | 1 place left | 0.347 |
| ÷ 100 | 2 places left | 0.0347 |
| ÷ 1000 | 3 places left | 0.00347 |
| × 0.1 | 1 place left | 0.347 |
| × 0.01 | 2 places left | 0.0347 |
| ÷ 0.1 | 1 place right | 34.7 |
| ÷ 0.01 | 2 places right | 347 |
Multiplying by 0.1 makes a number smaller, and dividing by 0.1 makes it bigger. That feels wrong, and it is still true: dividing by a tenth asks how many tenths fit, and ten of them fit in every one.
Worked example — 4.2 × 300. Forget the zeros and do 4.2 × 3 = 12.6, then hop the point two places right for the hundred: 1260.
Multiplication shortcuts
| Instead of | Do this | Example |
|---|---|---|
| × 4 | double, double | 23 × 4 = 46 → 92 |
| × 5 | × 10, halve | 86 × 5 = 860 → 430 |
| × 8 | double three times | 15 × 8 = 30 → 60 → 120 |
| × 9 | × 10, take the number off | 47 × 9 = 470 − 47 = 423 |
| × 11 | × 10, add the number | 47 × 11 = 470 + 47 = 517 |
| × 15 | × 10, add half again | 24 × 15 = 240 + 120 = 360 |
| × 25 | × 100, ÷ 4 | 36 × 25 = 3600 ÷ 4 = 900 |
| × 50 | × 100, halve | 86 × 50 = 8600 → 4300 |
| × 99 | × 100, take the number off | 46 × 99 = 4600 − 46 = 4554 |
| awkward pairs | double one, halve the other | 35 × 14 = 70 × 7 = 490 |
| a square ending in 5 | tens × next, then write 25 | 35² → 3 × 4 = 12 → 1225 |
| either side of a round number | round² − 1 | 21 × 19 = 400 − 1 = 399 |
And the one that works every time: split the number. 7 × 68 is 7 × 70 minus 7 × 2, which is 490 − 14 = 476.
Worked example — 24 × 25. Take the long way round: × 100 is 2400, and ÷ 4 is 600.
Division and the divisibility tests
These tell you whether a division will come out exactly — which is how you rule out three of the four options before dividing anything.
| Divides by | Test | Example |
|---|---|---|
| 2 | last digit is even | 3574 ✓ |
| 3 | digits add to a multiple of 3 | 1482 → 15 ✓ |
| 4 | last two digits divide by 4 | 7316 → 16 ✓ |
| 5 | ends in 0 or 5 | 2045 ✓ |
| 6 | passes both 2 and 3 | 1482 ✓ |
| 8 | last three digits divide by 8 | 9104 → 104 ✓ |
| 9 | digits add to a multiple of 9 | 5823 → 18 ✓ |
| 10 | ends in 0 | 4370 ✓ |
| 11 | alternate digits: the two sums differ by 0 or 11 | 4829 → (4+2) − (8+9) = −11 ✓ |
| 12 | passes both 3 and 4 | 2136 ✓ |
| 25 | ends 00, 25, 50 or 75 | 8175 ✓ |
Dividing by a decimal: hop both points the same number of places until the number you are dividing by is whole. 7.2 ÷ 0.08 becomes 720 ÷ 8 = 90.
Worked example — will 6 go into 4236? The last digit is even, and 4 + 2 + 3 + 6 = 15 divides by 3. It passes both, so yes — and 4236 ÷ 6 = 706.
Squares, cubes, powers and primes
| n | n² | n³ | 2ⁿ |
|---|---|---|---|
| 1 | 1 | 1 | 2 |
| 2 | 4 | 8 | 4 |
| 3 | 9 | 27 | 8 |
| 4 | 16 | 64 | 16 |
| 5 | 25 | 125 | 32 |
| 6 | 36 | 216 | 64 |
| 7 | 49 | 343 | 128 |
| 8 | 64 | 512 | 256 |
| 9 | 81 | 729 | 512 |
| 10 | 100 | 1000 | 1024 |
The squares past ten turn up constantly in area and in pattern questions.
| n | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|
| n² | 121 | 144 | 169 | 196 | 225 |
| n | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|
| n² | 256 | 289 | 324 | 361 | 400 |
The primes below 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Two is the only even one, and 1 is not prime at all — a prime has exactly two factors, and 1 has one.
Worked example — is 91 prime? It is not even, it does not end in 0 or 5, and its digits add to 10. But 7 × 13 = 91, so no. 7 is the one most people stop just before.
Angles on a clock face
A clock is a circle, so a whole turn is 360°, and twelve hour marks cut it into gaps of 30°. Everything else follows from that.
| Hand | Moves | Which is |
|---|---|---|
| minute hand | 360° in 60 minutes | 6° a minute, 30° every 5 minutes |
| hour hand | 360° in 12 hours | 30° an hour, ½° a minute |
The hour hand creeps. At half past three it is not on the 3 — it is halfway to the 4, which is where most clock questions are lost.
For any time, the angle is |30H − 5½M|, using the hour and the minutes. If the answer comes out over 180, take it off 360.
| Time | Working | Angle |
|---|---|---|
| 3:00 | 30×3 − 0 | 90° |
| 4:00 | 30×4 − 0 | 120° |
| 6:00 | 30×6 − 0 | 180° |
| 1:30 | 30 − 165 | 135° |
| 3:30 | 90 − 165 | 75° |
| 6:30 | 180 − 165 | 15° |
| 9:30 | 270 − 165 | 105° |
| 4:20 | 120 − 110 | 10° |
| 12:30 | 0 − 165 | 165° |
Worked example — the angle at 8:20. The hour part is 30 × 8 = 240 and the minute part is 5½ × 20 = 110, so |240 − 110| = 130°, which is already under 180.
Time zones and flights
A place that is ahead has a clock showing a later time than yours; a place that is behind shows an earlier one. Going east, clocks go forward. A question will always tell you the difference, so use the number it gives rather than one you half remember — Sydney is 8, 9 or 10 hours ahead of Paris depending on whose daylight saving is running, and the paper decides which.
Do it in this order and it comes out right every time.
| Step | What to do | Leaves Sydney 21:00, Paris 9 behind, 23 hour flight |
|---|---|---|
| 1 | put the take-off on the arrival clock | 21:00 − 9 = 12:00 in Paris |
| 2 | add the flight time | 12:00 + 23 = 35:00 |
| 3 | over 24? take 24 off and move the date on | 35 − 24 = 11:00 the next day |
Changing the clock first and adding second is what keeps it straight. Adding first means holding two sums at once, and that is where the hour goes missing.
| The question says | Do this to the departure time |
|---|---|
| the other place is n hours behind | take off n |
| the other place is n hours ahead | add n |
| you flew east | the clock goes forward |
| you flew west | the clock goes back |
| under 0 after the change | add 24, and it is the day before |
| 24 or over | take off 24, and it is the day after |
Asked for the length of a flight instead? Put both ends on the same clock first, then subtract. A plane leaving Sydney at 10:00 and landing in Perth at 13:00 with Perth 3 hours behind was in the air for 6 hours, not 3: 10:00 in Sydney is 07:00 in Perth, and 07:00 to 13:00 is six hours.
Worked example — a plane leaves Perth at 06:30 for a five hour flight, and Sydney is 3 hours ahead. Put the take-off on the Sydney clock first: 09:30. Then add the flight: it lands at 14:30.
Unit conversions
One rule covers all of them: going to a smaller unit multiplies, going to a bigger unit divides. The number of millimetres in something is always bigger than the number of metres, which is the check that catches a wrong hop.
| Length | Equals |
|---|---|
| 1 cm | 10 mm |
| 1 m | 100 cm = 1000 mm |
| 1 km | 1000 m = 100 000 cm |
| Mass | Equals |
|---|---|
| 1 g | 1000 mg |
| 1 kg | 1000 g |
| 1 tonne | 1000 kg |
| Capacity | Equals |
|---|---|
| 1 L | 1000 mL |
| 1 kL | 1000 L |
| 1 mL | 1 cm³ |
| 1 L | 1000 cm³ |
| 1 m³ | 1000 L = 1 kL |
Area and volume square and cube the hop, which is why 1 m² is 10 000 cm² and not 100.
| Area | Equals |
|---|---|
| 1 cm² | 100 mm² |
| 1 m² | 10 000 cm² |
| 1 hectare | 10 000 m² |
| 1 km² | 100 hectares = 1 000 000 m² |
| Time | Equals |
|---|---|
| 1 minute | 60 seconds |
| 1 hour | 60 minutes = 3600 seconds |
| 1 day | 24 hours = 1440 minutes |
| 1 week | 7 days |
| 1 year | 365 days (366 in a leap year) |
| 1 leap year | every 4th year |
Worked example — 2.5 km in centimetres. Down one step to metres: 2.5 × 1000 = 2500 m. Down again to centimetres: × 100 = 250 000 cm.
Perimeter, area and volume
| Shape | Area | Also |
|---|---|---|
| rectangle | length × width | perimeter = 2 × (l + w) |
| square | side² | perimeter = 4 × side |
| triangle | ½ × base × height | height is at right angles to the base |
| parallelogram | base × height | not the slanted side |
| trapezium | ½ × (a + b) × height | a and b are the parallel sides |
| circle | π × r² | circumference = π × d; π ≈ 3.14 or 22/7 |
| Solid | Volume | Also |
|---|---|---|
| cube | side³ | surface area = 6 × side² |
| rectangular prism | l × w × h | surface area = 2(lw + lh + wh) |
| any prism | area of the end × length | works for triangular prisms and cylinders |
Areas are in square units and volumes in cube units. An answer in cm where the question wanted cm² is marked wrong even when the number is right.
Worked example — a path one metre wide around a lawn 5 m by 3 m. The outside measures 7 m by 5 m, which is 35 m², and the lawn is 15 m², so the path is 20 m². Subtracting one rectangle from another beats trying to add up four strips and a corner.
Angles and shapes
| Fact | Adds to |
|---|---|
| angles on a straight line | 180° |
| angles around a point | 360° |
| angles in a triangle | 180° |
| angles in a quadrilateral | 360° |
| a right angle | 90° |
| a full turn | 360° |
Vertically opposite angles are equal. Where a line crosses two parallel lines, the angles in F shapes and Z shapes are equal, and the ones in a C shape add to 180°.
| Polygon | Sides | Angles add to | Each angle if regular |
|---|---|---|---|
| triangle | 3 | 180° | 60° |
| quadrilateral | 4 | 360° | 90° |
| pentagon | 5 | 540° | 108° |
| hexagon | 6 | 720° | 120° |
| heptagon | 7 | 900° | ≈ 128.6° |
| octagon | 8 | 1080° | 135° |
| decagon | 10 | 1440° | 144° |
The pattern is (n − 2) × 180 for the inside angles, and the outside angles of any regular shape always add to 360°, so each one is 360 ÷ n.
| Angle | Size |
|---|---|
| acute | less than 90° |
| right | exactly 90° |
| obtuse | between 90° and 180° |
| straight | exactly 180° |
| reflex | more than 180° |
Worked example — two angles of a triangle are 48° and 67°. They add to 115, and the three must make 180, so the third is 65°.
Shapes worth knowing by name
The four-sided shapes are the ones that cost marks, because three of them look alike until you know what to check.
| Shape | What makes it that shape | Lines of symmetry |
|---|---|---|
| square | four equal sides, four right angles | 4 |
| rectangle | opposite sides equal, four right angles | 2 |
| rhombus | four equal sides, no right angles needed | 2 |
| parallelogram | both pairs of opposite sides parallel | 0 |
| trapezium | exactly one pair of parallel sides | 0, or 1 if isosceles |
| kite | two pairs of equal sides, next to each other | 1 |
A square is a rhombus and a rectangle and a parallelogram all at once, which is why "is this a rhombus?" is answered by counting sides rather than by how it is drawn. In a rhombus and a kite the diagonals cross at right angles; in a rectangle they are equal.
| Solid | What it is | Faces, edges, corners |
|---|---|---|
| cube | six square faces | 6, 12, 8 |
| rectangular prism | a box | 6, 12, 8 |
| triangular prism | triangle ends, rectangle sides | 5, 9, 6 |
| square pyramid | square base, four triangles to a point | 5, 8, 5 |
| triangular pyramid | four triangles | 4, 6, 4 |
| cylinder | circle ends, no corners | 3 surfaces, 2 edges, 0 |
| cone | circle base, one point | 2 surfaces, 1 edge, 1 |
| sphere | a ball | 1 surface, 0, 0 |
A prism is the same shape all the way through, so slicing it anywhere gives the same face; a pyramid narrows to a point. Both are named after that end: a hexagonal prism has hexagon ends. For an end with n sides a prism has n + 2 faces, 3n edges and 2n corners, and a pyramid on an n-sided base has n + 1 faces, 2n edges and n + 1 corners.
Two kinds of symmetry, and questions ask for both. A line of symmetry folds the shape onto itself. Rotational symmetry is how many times it looks the same in one full turn — a rectangle is order 2, an equilateral triangle order 3, and any regular shape with n sides has n lines of symmetry and order n.
Worked example — a prism has 12 faces. Faces are n + 2, so its ends have 10 sides, and edges are 3n, which makes 30.
Circles, and what π is
Take any circle at all, measure the way round and the way across, and divide one by the other. You always get the same number, a little over three. That number is π — not a thing to picture, just the answer to that division, the same for a coin and for a planet.
| Part | What it is |
|---|---|
| centre | the middle point |
| radius | centre to the edge |
| diameter | right across, through the centre — two radiuses |
| circumference | the whole way round |
| chord | a straight line from edge to edge, not through the centre |
| arc | part of the way round |
| sector | a slice, like a piece of pie |
| semicircle | half; a quadrant is a quarter |
| Want | Formula | With r = 7, π = 22/7 |
|---|---|---|
| circumference | π × d, or 2 × π × r | 44 |
| area | π × r × r | 154 |
| diameter from radius | 2 × r | 14 |
| radius from diameter | d ÷ 2 | 7 |
Use 22/7 when the numbers in the question are multiples of 7, and 3.14 otherwise — that is why a question hands you a radius of 7 or 14 rather than 10.
Worked example — a circle of radius 10 cm, taking π as 3.14. The way round is 2 × 3.14 × 10 = 62.8 cm, and the area is 3.14 × 10 × 10 = 314 cm².
Speed, averages and rates
Draw the triangle in the margin before you start. Distance on top, speed and time underneath, and whichever one the question asks for, cover it with a finger: what is left is the sum.
| Want | Formula |
|---|---|
| speed | distance ÷ time |
| distance | speed × time |
| time | distance ÷ speed |
| average (mean) | total ÷ how many |
| the total | average × how many |
| unit price | price ÷ quantity |
The units have to match before you divide: 30 minutes is 0.5 hours, not 0.3.
| Average | What it is | Watch out for |
|---|---|---|
| mean | add them all, divide by how many | one odd value drags it |
| median | the middle one | put them in order first |
| mode | the most common one | there can be two, or none |
| range | biggest − smallest | it is a spread, not an average |
Average questions are nearly always total questions. If five tests average 72, the total is 360, and a sixth test of 90 makes 450 over six, which is 75.
Ratio: add the parts to get the number of shares. Sharing $180 in the ratio 4 : 5 is 9 shares of $20, so $80 and $100.
Two things moving at once is the question that looks hardest and is not. Work out how fast the gap between them changes, and then it is one journey at that speed.
| The two are | Gap speed | Then |
|---|---|---|
| coming towards each other | add the speeds | time = gap ÷ combined speed |
| going the same way, one catching up | subtract the speeds | time = gap ÷ difference |
| going opposite ways from one place | add the speeds | gap = combined speed × time |
| one left earlier | work out its head start first | then treat it as a catch-up |
One worked case for each line of that table, in the same order.
Coming towards each other — two cars start 300 km apart and drive towards each other at 60 and 40 km/h. The gap closes at 60 + 40 = 100 km/h, so they meet after 300 ÷ 100 = 3 hours. (The faster car has gone 180 km and the slower one 120 km, which is the 300 back again.)
Catching up — a car doing 60 km/h is 40 km ahead when another sets off after it at 80 km/h. The gap shrinks at 80 − 60 = 20 km/h, so it takes 40 ÷ 20 = 2 hours. Only the difference matters here: the same answer whether the two are doing 60 and 80 or 160 and 180.
Going opposite ways from one place — two cyclists leave the same park in opposite directions at 12 and 18 km/h. The gap grows at 12 + 18 = 30 km/h, so after two and a half hours they are 30 × 2.5 = 75 km apart.
One left earlier — a bus leaves at 9 am at 50 km/h and a car leaves the same stop at 10 am at 75 km/h. Work out the head start first: in that hour the bus went 50 km. Now it is a catch-up, gaining 75 − 50 = 25 km/h, so 50 ÷ 25 = 2 hours, and the car catches it at 12 noon. Check it: by then the bus has done 3 hours at 50 and the car 2 hours at 75, which is 150 km each.
And the trap in every average-speed question: a journey out at 60 and back at 40 does not average 50. Average speed is the whole distance over the whole time, so 120 km out and back takes 2 + 3 = 5 hours for 240 km, which is 48 km/h.
Worked example — a train leaves at 09:40 and arrives at 11:10, 120 km away. Cover S on the triangle: D ÷ T. The time is 1 hour 30 minutes, which is 1.5 hours, so 120 ÷ 1.5 = 80 km/h.
Getting the letter on its own
A letter is a number you do not know yet, and an equation is a pair of scales. Whatever you do to one side you do to the other, and it stays balanced.
The plan is always the same: letters to the left, plain numbers to the right, then divide.
| In the way | Do this to both sides | Example |
|---|---|---|
| + 7 | subtract 7 | x + 7 = 12 → x = 5 |
| − 4 | add 4 | x − 4 = 9 → x = 13 |
| × 3 | divide by 3 | 3x = 21 → x = 7 |
| ÷ 5 | multiply by 5 | x ÷ 5 = 4 → x = 20 |
| letters on both sides | take the smaller letter term off both | 5x + 2 = 3x + 14 |
| brackets | multiply them out first | 3(x + 2) = 18 → 3x + 6 = 18 |
That fifth row in full: 5x + 2 = 3x + 14. Take 3x off both sides for 2x + 2 = 14, take 2 off both for 2x = 12, divide both by 2, and x = 6.
Undo in the reverse order to the way it was built. If x was multiplied by 4 and then 3 added, take the 3 off first and divide by 4 second — the opposite order to the one the brackets rule uses going forwards.
Only like terms collect: 4x + 3x is 7x, but 4x + 3 is as far as 4x + 3 goes.
Worked example — 4(x − 3) = 20. Both sides divide by 4, giving x − 3 = 5, so x = 8. Dividing first was quicker than multiplying the brackets out.
Age problems
One fact does most of the work: everybody ages at the same rate, so the gap between two ages never changes. A father 26 years older than his son is 26 years older forever.
| The words | What to write |
|---|---|
| Sam is 4 years older than Pia | S = P + 4 |
| twice as old as | F = 2 × S |
| half as old as | S = F ÷ 2 |
| in 5 years | add 5 to both ages |
| 5 years ago | take 5 off both ages |
| their ages add to 40 | S + F = 40 |
Let the letter be the younger age; the other one is that plus the gap, and the sums stay small.
Then there is the shortcut worth knowing by heart. When one person is a whole number of times as old as another, the younger age at that moment is the gap divided by one less than the multiple.
| When | The younger one is | Gap of 24 |
|---|---|---|
| twice as old | the gap | 24 — so 24 and 48 |
| three times as old | gap ÷ 2 | 12 — so 12 and 36 |
| four times as old | gap ÷ 3 | 8 — so 8 and 32 |
| five times as old | gap ÷ 4 | 6 — so 6 and 30 |
So for "a father is 36 and his son is 10, when will the father be twice as old?": the gap is 26, the son will be 26, and that is in 16 years. Check it — 52 and 26 — rather than trusting it.
Worked example — Ben is 8 and his mother is 38. When will she be three times his age? The gap is 30 and never changes, so at three times, Ben is 30 ÷ 2 = 15 — in 7 years. Check: 15 and 45.
Factors, multiples, HCF and LCM
Factors go into a number; multiples come out of it. Twelve has factors 1, 2, 3, 4, 6 and 12 — six of them — and multiples 12, 24, 36 and on forever.
Find factors in pairs and you cannot miss one: 1 × 24, 2 × 12, 3 × 8, 4 × 6, and stop when the pair meets in the middle. A square number has an odd count of factors, because one pair is the same number twice.
| Asked for | What it means | How to get it |
|---|---|---|
| HCF | highest common factor | the biggest number that goes into both |
| LCM | lowest common multiple | the first number in both times tables |
For small numbers, list and look. For anything bigger, split both into primes: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. The HCF takes what they share — 2 × 2 × 3 = 12 — and the LCM takes the most of each — 2 × 2 × 2 × 3 × 3 = 72.
| Pair | HCF | LCM |
|---|---|---|
| 4 and 6 | 2 | 12 |
| 6 and 8 | 2 | 24 |
| 9 and 12 | 3 | 36 |
| 10 and 15 | 5 | 30 |
| 12 and 18 | 6 | 36 |
| 24 and 36 | 12 | 72 |
The check that catches a slip every time: HCF × LCM = the two numbers multiplied. For 12 and 18 that is 6 × 36 = 216, and 12 × 18 = 216 too.
Which one a question wants is in the story rather than in the words. Cutting something into equal pieces, or sharing into the largest equal groups, is HCF. Two things happening together again — buses, lights, laps — is LCM.
Worked example — two buses leave at 8 am, one every 12 minutes and one every 18. When do they next leave together? That is the LCM of 12 and 18, which is 36, so 8:36 am.
Probability
Probability is a fraction: how many ways the thing can happen, over how many ways anything can happen. Nothing more.
It runs from 0 to 1 and never leaves — 0 is impossible, 1 is certain, ½ is even chances — so an answer above 1 is a sum gone wrong rather than a very likely thing.
| Words | Means | Roughly |
|---|---|---|
| impossible | cannot happen | 0 |
| unlikely | probably not | under ½ |
| even chance | as likely as not | ½ |
| likely | probably | over ½ |
| certain | must happen | 1 |
| The question | The working | The answer |
|---|---|---|
| a coin lands heads | 1 of 2 | 1/2 |
| a dice shows 4 | 1 of 6 | 1/6 |
| a dice shows an even number | 3 of 6 | 1/2 |
| a dice shows more than 4 | 2 of 6 | 1/3 |
| a red ball from 3 red and 5 blue | 3 of 8 | 3/8 |
| not a red ball from those 8 | 5 of 8 | 5/8 |
The second-commonest trick is the one just above: the chance of something not happening is 1 take away the chance that it does. The commonest is a bag that gets smaller — take a ball out and keep it, and the next draw is out of seven, not eight.
For two things at once, count all the ways they can land. Two dice have 6 × 6 = 36 outcomes, and seven is the most likely total because six of those 36 make it.
Expected number is probability × how many tries: roll a dice 60 times and a four should come up about 1/6 × 60 = 10 times. About, not exactly — that is what makes it chance.
Worked example — a bag holds 4 red, 6 blue and 10 green counters. The chance of blue is 6 out of 20, which is 3/10. The chance of not blue is 7/10, and 3/10 + 7/10 = 1, which is the check.
Number facts worth memorising
Order of operations: brackets, then indices, then × and ÷ left to right, then + and − left to right.
| Idea | What to remember |
|---|---|
| rounding | look at the next digit only; 5 or more rounds up |
| negative numbers | subtracting a negative adds; two negatives multiply to a positive |
| factors | numbers that divide into it — they come in pairs |
| multiples | its times table, going on forever |
| HCF | the biggest factor both share |
| LCM | the first number in both times tables |
| square root | the side of that square: √144 = 12 |
| place value | tenths, hundredths, thousandths after the point |
Any number ÷ itself is 1, any number × 0 is 0, and anything to the power 0 is 1. A number is divisible by 6 only if it passes 2 and 3 — checking 6 directly is slower and less reliable.
Worked example — 6 + 2 × (9 − 4)² ÷ 5. Brackets first: 5. Then the index: 25. Then × and ÷ left to right: 2 × 25 = 50, ÷ 5 = 10. Then the addition: 16.
Adding up a run of numbers
Never add a run of numbers one at a time. A run that climbs in equal steps adds up to its middle value times how many there are, and every row below is that one idea wearing a different hat.
| To add | Do this | Example |
|---|---|---|
| 1 + 2 + … + n | n × (n + 1) ÷ 2 | 1 to 100 → 100 × 101 ÷ 2 = 5050 |
| any run of whole numbers | (first + last) ÷ 2 × how many | 21 + … + 30 → 25.5 × 10 = 255 |
| how many are in the run | last − first + 1 | 21 to 30 → 10 numbers |
| a run with an odd count | middle × how many | 17 + 18 + 19 → 18 × 3 = 54 |
| the first n odd numbers | n² | 1 + 3 + 5 + 7 + 9 = 5² = 25 |
| the first n even numbers | n × (n + 1) | 2 + 4 + 6 + 8 + 10 = 5 × 6 = 30 |
| multiples of k up to k × n | k × n(n + 1) ÷ 2 | 3 + 6 + … + 30 = 3 × 55 = 165 |
If the run steps by something other than 1 — 5, 10, 15, 20 … — the same two lines work. Count them with (last − first) ÷ step + 1, then multiply the average of the first and last by that count: 5 to 50 is 10 numbers, and 27.5 × 10 = 275.
Worked example — 2 + 4 + 6 + … + 40. There are 20 even numbers in it, halfway between 2 and 40 is 21, and 21 × 20 = 420.
Arrangements and choices
Two questions that sound the same and are not. Lining people up counts orders, so ABC and CBA are different. Picking a team does not, so they are the same team.
Factorials are the engine of both. n! means multiply every whole number from n down to 1.
| n | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| n! | 6 | 24 | 120 | 720 | 5040 | 40 320 |
| The question | Count it as |
|---|---|
| line up all n | n! = n × (n−1) × … × 1 |
| line up r of the n | ⁿPᵣ: count down from n, r numbers |
| choose r of the n | ⁿCᵣ = ⁿPᵣ ÷ r! |
| choose 2 of the n | ⁿC₂ = n × (n−1) ÷ 2 |
| seat n in a circle | (n−1)!, because one seat is fixed |
| r picks, repeats allowed | nʳ, n choices each time |
| a word with repeated letters | n! ÷ the repeats: LEVEL = 5! ÷ (2! × 2!) = 30 |
Written out in full, ⁿPᵣ = n! ÷ (n − r)! and ⁿCᵣ = n! ÷ (r! × (n − r)!). You will almost never want them in that form: counting down from n is the same sum with the cancelling already done, and 8 × 7 × 6 beats working out 8! and 5! and dividing one by the other.
The division in a combination is undoing a double count. Picking Ana then Ben and picking Ben then Ana are two arrangements of one pair, so the 2 arrangements of every pair are divided out. For threes it is 3! = 6, and for fours 4! = 24.
| Problem | Working | Answer |
|---|---|---|
| gold, silver, bronze from 8 runners | 8 × 7 × 6 | 336 |
| pick 3 of those 8 for a relay team | 336 ÷ 6 | 56 |
| arrange 5 books on a shelf | 5! | 120 |
| seat 5 people around a round table | 4! | 24 |
| handshakes among 10 people | 10 × 9 ÷ 2 | 45 |
| lines through 7 points, no 3 in a row | 7 × 6 ÷ 2 | 21 |
| 2 toppings chosen from 6 | 6 × 5 ÷ 2 | 15 |
| a 3-digit code, digits may repeat | 10 × 10 × 10 | 1000 |
| a 3-digit code, digits all different | 10 × 9 × 8 | 720 |
Worked example — how many ways can 4 of 7 books be lined up on a shelf? Order matters, so count down from 7 four times: 7 × 6 × 5 × 4 = 840.
Counting traps that cost marks
These appear in every selective paper, and each of them has a single fact that solves it.
| Trap | The fact |
|---|---|
| fence posts and trees | a fence with 10 gaps has 11 posts |
| pages, dates, seats: how many from 7 to 19? | last − first + 1 = 13, not 12 |
| cutting a log into 6 pieces | 5 cuts |
| a clock striking 6 in 5 seconds | the gaps are what take the time |
| working backwards | undo each step in reverse, swapping + for − |
Worked example — how many pages from page 45 to page 128? Last − first + 1 = 128 − 45 + 1 = 84. Subtracting alone would have lost the page you started on.
Worked examples
What is 0.0085 × 1000?
A thousand has three zeros, so the point hops three places right.
0.0085 → 0.085 → 0.85 → 8.5.
Answer: 8.5
Work out 35% of 180 without a calculator.
Build it: 10% of 180 is 18, so 30% is 54. Half of 18 is 9, which is the 5%.
54 + 9 = 63.
Answer: 63
What is the angle between the hands at 7:20?
Use |30H − 5½M|. The hour part is 30 × 7 = 210. The minute part is 5.5 × 20 = 110.
210 − 110 = 100, which is already under 180, so that is the answer.
Answer: 100°
A tank holds 2.4 m³. How many litres is that, and what does the water weigh?
1 m³ is 1000 L, so 2.4 m³ is 2400 L.
One litre of water weighs 1 kg, so 2400 L weighs 2400 kg, which is 2.4 tonnes.
Answer: 2400 L, weighing 2.4 tonnes
Five of Mia's spelling tests average 16 out of 20. She scores 19 on the sixth. What is her new average?
Averages are totals in disguise. Five tests averaging 16 is a total of 5 × 16 = 80.
80 + 19 = 99 over six tests.
99 ÷ 6 = 16.5.
Answer: 16.5
A regular hexagon and a square share a side. What is the angle between them at that corner?
Each angle of a regular hexagon is 120° and each angle of a square is 90°.
The three angles at that corner go all the way round, so they add to 360°.
360 − 120 − 90 = 150.
Answer: 150°
What is 31 + 32 + 33 + … + 50?
How many numbers: 50 − 31 + 1 = 20.
Halfway between the first and the last: (31 + 50) ÷ 2 = 40.5.
40.5 × 20 = 810.
Answer: 810
Eight friends each shake hands once with everybody else. How many handshakes?
Each person shakes 7 hands, which gives 8 × 7 = 56.
But you shaking my hand and me shaking yours is one handshake counted twice, so halve it.
Answer: 28
A club of 6 picks a president and a secretary. Then it picks 2 members for a trip. How many ways for each?
President and secretary are different jobs, so order matters: 6 × 5 = 30.
The two going on the trip are just two members, so order does not matter. Every pair was counted twice: 30 ÷ 2 = 15.
Answer: 30 ways, then 15 ways
A flight leaves Sydney at 21:40 and takes 23 hours 20 minutes. Paris is 9 hours behind Sydney. What time does it land in Paris?
Put the take-off on the Paris clock first: 21:40 − 9 hours = 12:40, the same day.
Now add the flight: 12:40 + 23 hours 20 minutes = 36:00.
That is past 24, so take 24 off and move the date on one.
Answer: 12:00 the next day
A mother is 32 and her daughter is 4. In how many years will the mother be three times as old as her daughter?
The gap is 28 years, and it never changes.
For three times as old, the younger one is the gap ÷ 2 = 14.
The daughter is 4 now, so that is in 10 years. Check: 42 and 14, and 3 × 14 = 42.
Answer: In 10 years
A prism has ends with seven sides. How many edges, faces and corners has it?
For an end with n sides: 3n edges, n + 2 faces, 2n corners.
So 21 edges, 9 faces and 14 corners.
Check with faces + corners − edges = 2: 9 + 14 − 21 = 2. ✓
Answer: 21 edges, 9 faces, 14 corners
A bicycle wheel has a diameter of 70 cm. How far does it travel in one full turn? Use π = 22/7.
One full turn covers the circumference, which is π × d.
22/7 × 70 = 22 × 10 = 220 cm.
The 70 is there to cancel the 7 — that is why the question picked it.
Answer: 220 cm, or 2.2 m
Solve 5x + 2 = 3x + 14.
Letters to the left: take 3x off both sides, leaving 2x + 2 = 14.
Numbers to the right: take 2 off both sides, leaving 2x = 12.
Divide both sides by 2.
Check: 5 × 6 + 2 = 32, and 3 × 6 + 14 = 32. ✓
Answer: x = 6
Two cyclists set off towards each other from towns 45 km apart, one at 14 km/h and the other at 16 km/h. How long until they meet?
They are closing the gap between them at 14 + 16 = 30 km/h.
The gap is 45 km, so cover T on the triangle: D ÷ S.
45 ÷ 30 = 1.5 hours.
Answer: 1 hour 30 minutes
What is the HCF and the LCM of 18 and 24?
Split both into primes: 18 = 2 × 3 × 3 and 24 = 2 × 2 × 2 × 3.
The HCF takes what they share: 2 × 3 = 6.
The LCM takes the most of each: 2 × 2 × 2 × 3 × 3 = 72.
Check: 6 × 72 = 432, and 18 × 24 = 432 too.
Answer: HCF 6, LCM 72
A bag has 5 red, 3 blue and 2 yellow marbles. One is taken out without looking. What is the chance it is not red?
There are 10 marbles altogether, and 5 of them are red.
The chance of red is 5/10, which is 1/2.
Not red is 1 − 1/2. Or count them: 3 + 2 = 5 of the 10 are not red, same answer.
Answer: 1/2
Fence posts go in every 3 metres along a straight 48 metre fence, with one at each end. How many posts?
48 ÷ 3 = 16, but that counts the gaps, not the posts.
A straight line of 16 gaps has 17 posts.
Answer: 17
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