AFM Enlistment (Maths) cheat sheet

Order of operations (BODMAS/PEMDAS): brackets, orders (powers/roots), division and multiplication left to right, then addition and subtraction left to right.

Multiplying or dividing two numbers with the same sign gives a positive result; opposite signs give a negative result; subtracting a negative is the same as adding its absolute value.

Divisibility tests: by 3 when digit sum is divisible by 3; by 9 when digit sum is divisible by 9; by 4 when the last two digits form a multiple of 4; by 8 when the last three digits do.

A number is prime if its only positive divisors are 1 and itself; 1 is not prime, and 2 is the only even prime.

Highest common factor is the product of shared prime factors at their lowest powers; lowest common multiple uses all prime factors at their highest powers.

Rounding: locate the required place value, then round up if the next digit is 5 or more, down otherwise; round only once, at the end of a calculation.

Multiplying by 10, 100 or 1000 shifts the decimal point right by the number of zeros; dividing shifts it left by the same count.

Estimate first by rounding to one significant figure to check whether a computed answer is of the right magnitude.

Equivalent fractions are made by multiplying or dividing numerator and denominator by the same non-zero number; this never changes the fraction's value.

To add or subtract fractions, rewrite them over a common denominator (the LCM of the denominators), then combine only the numerators.

To multiply fractions, multiply numerators together and denominators together; cancel common factors first to simplify.

To divide by a fraction, multiply by its reciprocal (flip numerator and denominator); a whole number n has reciprocal 1/n.

A mixed number is converted to an improper fraction by multiplying the whole part by the denominator and adding the numerator.

Comparing fractions: cross-multiply (a/b vs c/d compares ad with cb) or convert both to decimals.

A fraction is in simplest form when the HCF of numerator and denominator is 1; divide both by their HCF to reduce.

Of a quantity, a/b means divide the quantity by b and multiply by a; the word 'of' between fractions signals multiplication.

Percent means 'per hundred': p% = p/100; to convert a percentage to a decimal divide by 100, and to a fraction write it over 100 and simplify.

Percentage change = (new value − original value) ÷ original value × 100; a negative result is a decrease.

A percentage increase of p% multiplies the original by (1 + p/100); a decrease of p% multiplies by (1 − p/100).

To find the original amount after a percentage change, divide by the multiplier: if 120 is the result of a 20% increase, the original is 120 ÷ 1.2 = 100.

Repeated percentage changes multiply: two successive 10% increases give a factor of 1.1 × 1.1 = 1.21, i.e. a 21% total increase, not 20%.

A ratio a : b shares a total into a + b parts; each part equals total ÷ (a + b), then multiply by a and b.

Two quantities are in direct proportion when their ratio is constant; they are inversely proportional when their product is constant.

To express one quantity as a percentage of another, divide the first by the second and multiply by 100.

Collect like terms by adding or subtracting coefficients of identical variable parts; 3x and 5x combine to 8x, but 3x and 5x² do not.

Expand brackets by multiplying each term inside by the factor outside: a(b + c) = ab + ac, and (x + p)(x + q) = x² + (p + q)x + pq.

Solve a linear equation by performing the same operation on both sides until the variable is isolated; undo addition/subtraction first, then multiplication/division.

When an unknown appears on both sides, subtract the smaller variable term from both sides to gather it on one side.

Substitute into a formula by replacing each letter with its value in brackets, then evaluate using order of operations.

Rearrange a formula by applying inverse operations to both sides to make a chosen variable the subject.

An inequality reverses direction when both sides are multiplied or divided by a negative number.

Check a solution by substituting it back into the original equation; both sides must be equal.

Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal.

Interior angles of any triangle sum to 180°; the exterior angle equals the sum of the two opposite interior angles.

Interior angles of a polygon with n sides sum to (n − 2) × 180°; for a regular polygon each interior angle is that sum divided by n.

Area formulas: rectangle l × w, triangle ½ × base × height, parallelogram base × height, trapezium ½(a + b)h, circle πr².

Circumference of a circle is πd or 2πr; use π ≈ 3.14 or the calculator's π key, and keep units consistent.

Pythagoras' theorem: in a right-angled triangle a² + b² = c², where c is the hypotenuse; rearrange to find a shorter side.

Volume: cuboid l × w × h, prism area of cross-section × length, cylinder πr²h; surface area of a cuboid is 2(lw + lh + wh).

Similar shapes have equal corresponding angles and proportional sides; if the length scale factor is k, area scales by k² and volume by k³.

Metric conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm, 1 kg = 1000 g, 1 litre = 1000 ml; convert by multiplying or dividing by the relevant power of 10.

Speed = distance ÷ time; rearrange to distance = speed × time and time = distance ÷ speed, matching units (e.g. km/h with hours).

Mean = sum of values ÷ number of values; median = middle value of an ordered list; mode = most frequent value; range = highest − lowest.

With an even count of data values, the median is the mean of the two middle values after ordering.

To read a scale on a graph, first find the value of one small division by dividing the labelled interval by the number of divisions.

A pictogram or bar chart is read by multiplying the symbol or bar height by the stated key or scale value.

Density = mass ÷ volume; pressure = force ÷ area; use consistent metric units before calculating.

Average speed over a journey equals total distance divided by total time, not the mean of the separate speeds.