PSLCE Mathematics cheat sheet

Place value: in 47 382 the digit 7 stands for 7 000 because it is in the thousands column.

Reading large numbers in groups of three: 2 506 041 is 'two million, five hundred and six thousand and forty-one'.

Rounding rule: look at the digit immediately right of the target place; 5 or more rounds up, 4 or less stays.

Roman numerals use I=1, V=5, X=10, L=50, C=100, D=500, M=1000; a smaller symbol before a larger one subtracts (IV=4, IX=9, XL=40, XC=90).

A number is divisible by 3 if its digit sum is divisible by 3; by 9 if the digit sum is divisible by 9; by 4 if the last two digits form a multiple of 4.

Prime numbers have exactly two factors: 1 is not prime, 2 is the only even prime.

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144; their factors come in pairs except the square root.

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36; multiples of 6 begin 6, 12, 18, 24 and continue without end.

Ordering decimals and whole numbers: compare the highest place value first, working left to right.

Order of operations: Brackets, then Orders (powers/roots), then Division and Multiplication left to right, then Addition and Subtraction left to right.

Addition and multiplication are commutative (a+b=b+a, a×b=b×a); subtraction and division are not.

Multiplication by powers of 10 shifts digits left: ×10, ×100, ×1000 add 1, 2, 3 zeros respectively.

Long multiplication check: 234×56 = 234×50 + 234×6 = 11 700 + 1 404 = 13 104.

Division algorithm: divide, multiply, subtract, bring down the next digit, repeat; record any remainder.

Estimate before computing: 498×21 ≈ 500×20 = 10 000, so an answer near 10 458 is reasonable.

Inverse operations undo each other: if 47+58=105 then 105−58=47; if 12×9=108 then 108÷9=12.

A remainder must always be smaller than the divisor; dividing by zero is undefined.

Equivalent fractions are made by multiplying or dividing numerator and denominator by the same non-zero number: 3/4 = 9/12.

To add or subtract fractions, first write them over a common denominator, then combine only the numerators.

Multiply fractions straight across: 2/3 × 4/5 = 8/15.

Divide by a fraction by multiplying by its reciprocal: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.

A fraction is in simplest form when the only common factor of numerator and denominator is 1.

To compare fractions, cross-multiply or convert to decimals: 5/8 = 0.625 and 2/3 ≈ 0.667, so 2/3 is larger.

Mixed number to improper fraction: 3 1/2 = (3×2+1)/2 = 7/2.

Of means multiply: 3/5 of 40 = 3/5 × 40 = 24.

Decimal place values after the point: tenths, hundredths, thousandths, so 0.407 = 4 tenths + 0 hundredths + 7 thousandths.

Multiplying a decimal by 10, 100, 1000 moves the point 1, 2, 3 places right; dividing moves it left.

To convert a fraction to a decimal, divide the numerator by the denominator: 7/8 = 0.875.

Percent means 'per hundred': 35% = 35/100 = 0.35.

Percentage of a quantity: 15% of 240 = 0.15 × 240 = 36.

To express one quantity as a percentage of another: part ÷ whole × 100, e.g. 18/45 × 100 = 40%.

Percentage increase or decrease = change ÷ original × 100.

Rounding a decimal to 2 places: look at the third decimal digit; 3.476 rounds to 3.48.

Length: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km.

Mass: 1000 g = 1 kg, 1000 kg = 1 tonne (metric ton).

Capacity: 1000 mL = 1 L; 1 L of water has a mass of about 1 kg.

Area units are squared: 1 m² = 10 000 cm²; volume units are cubed: 1 m³ = 1 000 000 cm³.

Perimeter of a rectangle = 2(l + w); area of a rectangle = l × w; area of a triangle = ½ × base × height.

Volume of a cuboid = length × width × height.

To convert between units, multiply when going to a smaller unit and divide when going to a larger unit: 3.5 km = 3500 m.

Angles on a straight line add to 180°; angles at a point add to 360°.

Vertically opposite angles are equal; alternate and corresponding angles are equal when a transversal crosses parallel lines.

Triangle angle sum = 180°; quadrilateral angle sum = 360°.

Equilateral triangle: all sides equal, all angles 60°. Isosceles triangle: two equal sides and two equal base angles.

A square, rectangle, parallelogram and rhombus each have angle sum 360°; a parallelogram has opposite angles equal.

Circle: circumference = πd = 2πr; area = πr² (use π ≈ 3.14 or 22/7).

Congruent shapes are identical in size and shape; similar shapes have equal angles and proportional sides.

A line of symmetry divides a shape into two mirror-image halves; a square has 4, an equilateral triangle has 3.

Write money to two decimal places: 7 dollars 5 cents = $7.05, not $7.5.

Adding money: align the decimal points and keep two decimal places in the answer.

Change = amount paid − total cost; check by adding change back to the cost.

Unit price = total cost ÷ number of items; compare unit prices to find the better buy.

Profit = selling price − cost price; loss occurs when selling price is below cost price.

Simple interest = principal × rate × time (rate as a decimal per year).

Discount = percentage × marked price; sale price = marked price − discount.

Budget total = sum of all listed expenses; balance left = income − total expenses.

1 hour = 60 minutes, 1 minute = 60 seconds, 1 day = 24 hours, 1 week = 7 days.

1 year = 12 months = 365 days (366 in a leap year); a leap year is divisible by 4, century years by 400.

Months with 31 days: January, March, May, July, August, October, December; February has 28 or 29.

24-hour clock: add 12 to afternoon/evening times (3:45 p.m. = 15:45); midnight is 00:00 and noon is 12:00.

Time interval = end time − start time; borrow 60 minutes when subtracting, e.g. 10:15 − 9:40 = 35 minutes.

Average speed = distance ÷ time; distance = speed × time; time = distance ÷ speed.

To read a timetable, locate the departure row, then the arrival column; the elapsed time is the difference.

Duration in minutes = (hours difference × 60) + minutes difference, adjusting when minutes are negative.

Mean = sum of all values ÷ number of values.

Median = middle value when data are arranged in order; with an even count, average the two middle values.

Mode = the value that occurs most often; a data set may have one mode, several, or none.

Range = largest value − smallest value; it measures spread, not centre.

In a bar chart, read the value from the top of each bar against the scale; bars have equal width and gaps.

In a pictogram, multiply the number of symbols by the key value, including half symbols.

Pie chart: each sector angle = (frequency ÷ total) × 360°.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; probabilities range from 0 to 1.

Find a sequence rule by comparing consecutive terms: 5, 9, 13, 17 adds 4 each time, so the nth term is 4n + 1.

The nth term of a linear sequence is dn + c, where d is the common difference and c is found by substituting n = 1.

Square-number pattern 1, 4, 9, 16, 25 has nth term n²; triangular numbers 1, 3, 6, 10 have nth term n(n+1)/2.

Collect like terms: 3a + 5b − a + 2b = 2a + 7b; only terms with identical letters combine.

Solve an equation by doing the same operation to both sides: 4x + 7 = 31 gives 4x = 24, so x = 3.

Substitute carefully: if y = 3x − 2 and x = 5, then y = 15 − 2 = 13.

Use a letter for the unknown, write an equation from the words, solve it, then check by substitution.

Multi-step problems: identify what is asked, list the given facts, choose the operation, compute, and check the answer is sensible.