ICAS Mathematics cheat sheet

Order of operations: brackets, then indices, then multiplication and division (left to right), then addition and subtraction (left to right).

A whole number is divisible by 3 or 9 exactly when its digit sum is divisible by 3 or 9; by 4 when its last two digits form a multiple of 4; by 6 when it is divisible by both 2 and 3.

x% of N = (x ÷ 100) × N, and percentage change = (new − original) ÷ original × 100%.

To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.

For any two positive integers, HCF × LCM = the product of the numbers, so LCM = product ÷ HCF.

1 is not prime; 2 is the only even prime; every whole number greater than 1 has a unique prime factorisation.

Rounding to a given place value: inspect the next digit to the right - 5 or more rounds up, 4 or less stays the same.

Multiplying or dividing two numbers with the same sign gives a positive result; opposite signs give a negative result.

Sharing an amount in the ratio a : b divides it into a + b equal parts, with shares of a/(a+b) and b/(a+b) of the total.

An arithmetic sequence with first term a and constant difference d has nth term a + (n − 1)d.

In y = mx + c, m is the gradient and c is the y-intercept (the y-value when x = 0).

(x + a)(x + b) = x² + (a + b)x + ab; (a + b)² = a² + 2ab + b²; (a − b)(a + b) = a² − b².

Only like terms combine: 3x + 5x = 8x, but 3x and 3x² cannot be added because the powers differ.

When solving a linear equation, perform the same operation on both sides and undo operations in reverse order.

Constant first differences between consecutive terms indicate a linear rule; constant second differences indicate a quadratic rule.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Substituting into an expression: replace each variable with its value and keep brackets, e.g. 3(x − 2) with x = 5 gives 9.

Rectangle A = lw; triangle A = ½bh (b and h perpendicular); parallelogram A = bh; trapezium A = ½(a + b)h; circle A = πr².

Circle circumference C = 2πr = πd, with π ≈ 3.14159.

Prism volume = area of the uniform cross-section × length; for a cylinder V = πr²h.

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, triangle angles sum to 180°, quadrilateral angles sum to 360°.

An exterior angle of a triangle equals the sum of the two interior opposite angles.

Pythagoras' theorem: in a right-angled triangle a² + b² = c², where c is the hypotenuse opposite the right angle.

Length conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm; area conversions use squares, so 1 m² = 10 000 cm² and 1 cm² = 100 mm².

For similar figures, corresponding angles are equal, sides share a ratio k, areas scale by k² and volumes by k³.

Speed = distance ÷ time, so distance = speed × time and time = distance ÷ speed.

Mean = sum of values ÷ number of values; median = middle value of ordered data (average of the two middle values if the count is even); mode = most frequent; range = maximum − minimum.

Adding a constant k to every data value adds k to the mean and median but leaves the range and standard deviation unchanged.

Probability of an event = number of favourable outcomes ÷ total number of equally likely outcomes, so 0 ≤ P ≤ 1.

Complement rule: P(not A) = 1 − P(A).

Mutually exclusive events satisfy P(A or B) = P(A) + P(B); independent events satisfy P(A and B) = P(A) × P(B).

Rolling two six-sided dice gives 6 × 6 = 36 equally likely ordered outcomes.

In a pie chart, sector angle = (category frequency ÷ total frequency) × 360°.

The mean is more affected by outliers than the median, so the median better describes the centre of skewed data.