LC Foundation Maths cheat sheet

Order of operations: evaluate brackets first, then indices, then multiplication and division from left to right, then addition and subtraction from left to right.

For fractions: a/b + c/d = (ad + bc)/(bd); a/b ÷ c/d = a/b × d/c; simplify by cancelling common factors.

Percentage change = (new value − old value)/old value × 100%; an increase of p% multiplies by (1 + p/100), and a decrease of p% multiplies by (1 − p/100).

Standard form writes a number as a × 10^n, where 1 ≤ a < 10 and n is an integer; for example, 0.0042 = 4.2 × 10^−3.

HCF is the product of common prime factors using the lowest power of each; LCM is the product of all prime factors using the highest power of each.

Negative and fractional indices: a^−n = 1/a^n, a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a^m.

Rounding: locate the required place value; if the next digit is 5 or more, round up, otherwise keep the digit unchanged.

Bounds for a value rounded to the nearest unit: lower bound = value − 0.5 unit, upper bound = value + 0.5 unit; use half the rounding unit for other accuracies.

Expand two brackets: (x + a)(x + b) = x² + (a + b)x + ab; and (x + a)(x − a) = x² − a².

For x² + bx + c, factorise by finding p and q with pq = c and p + q = b, giving (x + p)(x + q).

Quadratic formula: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)]/(2a).

A straight line y = mx + c has gradient m and y-intercept c; gradient m = (y₂ − y₁)/(x₂ − x₁).

Index laws: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1.

Solve simultaneous equations by elimination: multiply equations to make coefficients equal and opposite, then add or subtract to eliminate one variable.

Rearranging formulas: perform the same inverse operation on both sides, undoing addition or subtraction before multiplication or division.

Function notation: f(x) means substitute x into the rule; for example, if f(x) = 2x + 1, then f(3) = 7.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.

To divide a quantity in the ratio a : b, total parts = a + b; one part = quantity/(a + b), then the shares are a parts and b parts.

A ratio compares like quantities; convert to the same units before simplifying, and write it in the form a : b with whole numbers where possible.

Direct proportion: y = kx, so y/x is constant; if x doubles, y doubles.

Inverse proportion: y = k/x, so xy is constant; if x doubles, y halves.

Speed = distance ÷ time; distance = speed × time; time = distance ÷ speed. Keep units consistent.

Density = mass ÷ volume; pressure = force ÷ area; population density = population ÷ area.

Percentage profit or loss = (profit or loss ÷ original cost) × 100%; selling price = cost ± profit or loss.

Compound interest: A = P(1 + r)^n, where P is principal, r is the decimal rate per period and n is the number of periods.

Map scale 1 : n means actual length = map length × n; convert map and actual lengths to the same unit before calculating.

Angle sums: angles on a straight line add to 180°; angles at a point add to 360°; angles in a triangle add to 180°; angles in a quadrilateral add to 360°.

Parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°.

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

Trigonometry: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Circle: circumference = 2πr; area = πr².

Sector: arc length = (θ/360) × 2πr; area = (θ/360) × πr², where θ is the central angle in degrees.

Area: triangle = ½bh; parallelogram = bh; trapezium = ½(a + b)h.

Similar shapes with length scale factor k: areas scale by k² and volumes scale by k³.

Bearings are measured clockwise from north and written as three figures, for example 045°.

Mean = sum of values ÷ number of values; for a frequency table, mean = Σfx ÷ Σf.

Median: arrange data in order; for n values, the median is the ((n + 1)/2)th value, or the mean of the two middle values when n is even.

Mode = most frequent value or category; range = highest value − lowest value.

Lower quartile Q1 and upper quartile Q3 split ordered data into quarters; interquartile range = Q3 − Q1.

Pie chart angle for a category = (frequency ÷ total frequency) × 360°.

Histograms show continuous data with bars whose areas represent frequencies; bar charts show frequencies by category with equal bar widths.

Scatter graph correlation: positive means both variables increase together, negative means one increases as the other decreases; correlation does not prove causation.

Probability = number of favourable outcomes ÷ total number of equally likely outcomes; 0 ≤ P ≤ 1.

Complement: P(not A) = 1 − P(A).

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

Tree diagrams: multiply probabilities along a branch to find an outcome's probability; add probabilities of branches for all outcomes satisfying the event.

Relative frequency = number of times the event occurs ÷ total number of trials; it estimates probability and usually stabilises with more trials.

A sample space lists all possible outcomes; for two independent choices, multiply the number of options to count outcomes.

With replacement, probabilities stay the same; without replacement, update the numerator and denominator after each selection.